Full-back Leigh Halfpenny missed two easy first-half penalties that looked to have derailed Welsh hopes of a first victory over the Wallabies since 2008.
But Halfpenny, the third-highest scorer in Welsh international rugby history behind Neil Jenkins and Stephen Jones with more than 700 points, managed two successful strikes off the tee before Biggar came up trumps.
Halfpenny was off the pitch, having been felled by a high tackle, which meant Biggar assumed kicking duties as he edged Wales home after Bernard Foley and Matt Toomua kicked penalties for Australia.
The Wallables’ defeat will now turn up the heat on head coach Michael Cheika as they suffered an eighth loss from their last 10 matches.
It was a game that will not live long in the memory, one that was littered with errors, yet Wales will not care one iota.
In addition to ending the losing sequence, they also made it seven wins on the bounce against all opponents for the first time since 2004/05 and landed a psychological blow before the countries meet again as World Cup pool rivals in Tokyo next September.
Wales showed three changes from the side that beat Scotland last weekend, with head coach Warren Gatland handing starts to wing Josh Adams, prop Tomas Francis and lock Adam Beard, while British & Irish Lions backs Biggar and Liam Williams featured among a strong replacements’ bench.
The opening 10 minutes saw both sides probing for an attacking opportunity, with Wales fly-half Gareth Anscombe making a brilliant cover tackle on Wallabies centre Samu Kerevi before Anscombe’s clever attacking kick into space only narrowly failed to find a sprinting Adams.
Centre Jonathan Davies also tested Australia defensively before Wales missed a chance to go ahead when Halfpenny uncharacteristically drifted a short-range penalty chance wide.
It was a let-off for the Wallabies, but Wales then gained a second penalty, and Halfpenny this time made no mistake, opening the scoring after 22 minutes.
Neither side could get a consistent upper hand, but Australia drew level seven minutes before the break when fly-half Foley kicked a 40-metre penalty after Wales flanker Dan Lydiate drifted offside.
Referee Ben O’Keeffe, though, completely missed a high arm-led challenge by Wales captain Alun Wyn Jones on Foley just before the penalty award, although it could yet attract further scrutiny.
It was the cue for Australia to step up a gear in terms of their attacking game, moving possession wide at pace, and Wales were stretched defensively before an Anscombe penalty to touch took them back inside the Wallabies’ half.
And Halfpenny then missed another sitter, putting the ball wide from in front of the posts and a scrappy first half ended 3-3.
Wales wing George North limped off early in the second half, with Williams replacing him, while Dillon Lewis was a straight substitution for Francis.
Australia then sacrificed a kickable penalty just 20 metres out, kicking for touch and an attacking lineout instead, but they knocked on possession and wasted the chance as Wales regrouped to clear danger.
But the Wallabies were growing in confidence, with Israel Folau starting to roam dangerously off his wing, and Wales required a lengthy Davies clearance to gain a foothold back in Australian territory as further substitutions saw runs for scrum-half Tomos Williams and prop Rob Evans.
And approaching the final quarter of a dour encounter, it remained all-square and crying out for a flash of inspiration.
Then came the late drama, though, and Biggar appeared off the bench to nudge Wales back in front after Toomua’s equalising strike two minutes earlier.
Head coach Eddie Jones refused to be drawn on the controversial decision which denied England a memorable win over New Zealand at Twickenham on Saturday.
England were trailing 16-15 after 76 minutes when Courtney Lawes charged down TJ Perenara’s kick and Sam Underhill scooped up the ball before surging to a solo try.
However, South African Television Match Official Marius Jonker and French referee Jerome Garces reviewed the incident and decided Lawes was offside.
Jones said: “I don’t comment on those decisions. I’ll leave it up to that guy. If he can’t take the right decision with 10 replays, who can?
“Sometimes the game loves you and sometimes the game doesn’t love you. We’ll get some love from the game further down the track.”
New Zealand head coach Steve Hansen said: “There was no doubt he’s offside. What was going through my mind was are they going to be brave enough to make the right decision? And they were.”
Underhill told BBC Radio 5 Live: “I don’t know what I was thinking when I went over.
“I wasn’t watching the screen. At times like that we don’t watch the decision, we look to the next job.
“When it got brought back it was what it was but it’s those small margins.”
It was a first meeting between England and New Zealand, winners of the last two World Cups, in four years and many expected the All Blacks to trounce their injury-depleted hosts.
But afterwards it was Jones who insisted England were on course for World Cup glory after the agonising defeat.
Tries from Chris Ashton and Dylan Hartley helped England to a 15-0 lead after 25 minutes, which the All Blacks eroded with a flourish at the end of the first half.
Beauden Barrett’s penalty saw New Zealand take the lead for the first time in the game after 60 minutes and they did not relinquish it.
Jones accentuated the positives and looked ahead to next year’s World Cup in Japan.
“We’re disappointed, but we’re excited about where we’re going,” Jones added.
“We’ll learn a lot from that today. We had opportunities to win the game, we didn’t take them, they did. They deserved to win the game.
“It’s a really good step forward. You benchmark yourself against New Zealand. New Zealand are the best team in the world.
“We’ve got to fix the things that didn’t work today. If we do that, we’re on the road to being the best team in the world, which is what we set out to be.”
Jones felt England were in the ascendancy in the final quarter, when a drop goal opportunity failed to materialise.
“I thought we played the final 20 (minutes) exceptionally well,” Jones added.
“That’s where New Zealand generally run away from teams and they couldn’t. They couldn’t break us.
“If we’d kept going for another five minutes, we maybe would’ve got them.”
Hooker and co-captain Hartley went off at half-time with a thumb injury he has been carrying, while lock George Kruis has a calf problem which will be assessed.
New Zealand’s influential centre Sonny Bill Williams went off with a groin injury in the first half.
The All Blacks won the World Cup on their most recent visit to Twickenham, in October 2015, when England became the first hosts to exit at the pool stage. They have been transformed under Jones, who was appointed in December 2015.
Hansen said: “He’s doing a good job here for England. Last time we were here was 2015 and things didn’t go that well for England.
“Since then they’ve won 18 in a row, they’re in the right place.
“They’ve showed today they’re going to be competitive against whoever you want to play against them.”
Hansen, though, had words of praise for his side too.
He added: “I thought England were very, very good and I thought we showed a lot of character.”
Courtney Lawes charged down TJ Perenara’s 76th-minute box kick and the ball fell to Underhill, who weaved full-back Damian McKenzie inside out to touch down in the left corner.
After reviewing the try with the TMO, however, referee Jerome Garces judged that Lawes had been in an offside position and awarded the All Blacks a penalty.
It was a marginal call and England responded with a late onslaught that failed to produce the crucial winning score, but it was a fine performance from Eddie Jones’ men that will serve them well heading into next year’s World Cup.
They led from the second minute until the 60th as early tries by Chris Ashton and Dylan Hartley helped them to a scarcely-believable 15-0 lead that was gradually eroded.
Beauden Barrett nudged the All Blacks in front with a penalty as England’s play grew progressively worse having set off like a steam train, their forwards bristling with aggression.
The momentum shift began late in the first half and was confirmed by a terrific try from McKenzie.
All Blacks coach Steve Hansen had raised expectations by declaring the match to be bigger than last year’s series against the British & Irish Lions, but an arm wrestle of a contest failed to provide quite the same drama.
England were depleted by injury but buoyed by a rousing 12-11 victory over South Africa a week earlier and they traded on equal terms with the world champions from start to finish.
Jones called on his players to act as movie directors rather than extras in the 41st instalment of the fixture and they did exactly that in the first half.
Strong carries from Underhill, Kyle Sinckler and Maro Itoje in a rampaging start sucked in defenders and when Ben Youngs spotted an unmarked Ashton stationed on the touchline, he sent the Sale wing sliding over with a floated pass.
McKenzie was shoved into touch by Underhill and then Jonny May as the All Blacks came under renewed pressure and England extended their lead with a drop goal by Owen Farrell.
Elliot Daly and then May running at speed into heavy traffic produced a roar from Twickenham, but it was in response to Hartley’s 24th minute-try that the crowd really erupted.
A lineout 15 metres out developed into an unstoppable maul that arrowed diagonally towards the posts and with backs adding their weight, a pile of white shirts crashed over the line with Hartley scoring.
New Zealand-born Brad Shields disappeared to the blood bin – an England player could be heard complaining on the ref mic that the flanker had been punched – and in a blow to the tourists Sonny Bill Williams followed him off the pitch with a groin problem.
The All Blacks strung together 23 phases and through Rieko Ioane they finally threatened the try line, the breakthrough arriving soon after when McKenzie ran a brilliant line to take an inside scoring pass from Barrett.
A Barrett penalty slashed the deficit to five points at half-time and when the teams emerged for the second-half Hartley had been replaced by Jamie George.
Ardie Savea dropped a poor pass by Aaron Smith with the line beckoning but a Barrett drop-goal – the first of his 71-cap career – continued to drain England’s lead.
Youngs almost dummied his way over from an attacking scrum as play swung to the opposite and New Zealand took the lead for the first time in the 60th minute through a Barrett penalty.
An ascendant scrum offered England a platform but they failed to make the most of it and with the clock ticking down Underhill’s heart was broken by the intervention of the officials after he had outfoxed McKenzie.