It marked a sixth game unbeaten for Everton under former England boss Allardyce, but Chelsea’s failure to convert possession into goals will not have pleased Blues manager Antonio Conte.
Conte has already conceded his inconsistent team won’t be able to defend the Premier League title and this was another frustrating afternoon for the Italian as the Blues squandered their territorial dominance.
Chelsea remain third and are 13 points behind leaders Manchester City, who play Bournemouth later on Saturday.
Initially, it looked like being an encouraging day for Conte as Chelsea quickly established a pattern of complete dominance.
📋 | Read our account of today’s stalemate at Goodison as the Blues notched clean sheet number 5️⃣ in their last 7️⃣ games…
— Everton (@Everton) December 23, 2017
It took an impressive double goal-line clearance by veteran defender Phil Jagielka to deny the visitors what would have been a deserved lead.
The Everton captain read the situation perfectly as Marcos Alonso’s cross flew through the home area after 10 minutes, reacting sharply to block Tiemoue Bakayoko’s effort.
Jagielka then responded even more impressively as Willian looked certain to convert the rebound, the defender positioning himself to keep out the follow-up effort in similar fashion.
Yet for all their superiority in those opening stages, Chelsea struggled to carve out many meaningful chances to trouble home keeper Jordan Pickford.
After 20 minutes, Cuco Martina was booked for a blatant foul on Victor Moses on the edge of the Everton area but Willian sent his free-kick sailing well over.
And while the passing and movement of Willian, Pedro and Eden Hazard was pleasing on the eye and threatening at times, Everton stuck to their defensive duties well.
It was easy to see the impact Allardyce has made since his arrival at Goodison Park although Everton were not helped by the pre-match loss of in-form Wayne Rooney due to illness.
In the absence of the 10-goal forward, Everton’s attacking play was likely to be hindered and so it proved, with Chelsea keeper Thibaut Courtois a virtual spectator.
A promising counter-attack, just after the half-hour, ended with Morgan Schneiderlin sending away Tom Davies but his strike from 25 yards was always drifting away from goal.
And before the interval, Dominic Calvert-Lewin showed great persistence to strip Andreas Christiansen of the ball and advance into the area where his shot was blocked by Antonio Rudiger.
But Chelsea responded in kind when Bakayoko broke up an Everton attack on the halfway line before playing Pedro clear for a powerful shot which drew a spectacular one-handed save from Pickford.
Allardyce made a major reshuffle at the interval, bringing on Sandro and Ashley Williams as he switched to three central defenders, a move that proved something of a gamble when he lost Idrissa Gueye to injury within five minutes of the restart and was forced to use his final substitute.
By then, Chelsea had already threatened Pickford’s goal, with the keeper saving well from Pedro’s drive before Williams did well to block Alonso’s effort.
— Everton (@Everton) December 23, 2017
There was some anxiety for the visitors when Hazard was hacked down and required lengthy treatment, with Michael Keane booked for the foul.
But the Belgian quickly shook off the effects of the injury and was soon forcing Pickford into another fine diving save after receiving a pass from Willian.
Williams got his head to Victor Moses’ fierce 75th minute cross, deflecting it onto the Everton bar before it flew out of harm’s way.
There was also a late chance for substitute Cesc Fabregas when Jagielka stuck out a foot and deflected his powerful, long-range shot just wide.
Arsenal’s clash with Liverpool on Friday night is the latest in a long line of exciting, high-scoring games to take place over the festive period.
Here, Press Association Sport looks at five other Christmas crackers.
Christmas Day 1957: Chelsea 7 Portsmouth 4
The last full programme of Christmas Day fixtures took place on 1957, and was notable for the performance of a 17-year-old called Jimmy Greaves.
Greaves scored his first league hat-trick as Chelsea thrashed Portsmouth 7-4 at Stamford Bridge.
Boxing Day 1963: Fulham 10 Ipswich 1
On a day when the 10 top-flight matches yielded an incredible 66 goals, 11 were scored at Craven Cottage, where Fulham ran riot against an Ipswich team who had been crowned league champions just 18 months earlier.
Even more remarkably, Ipswich would recover to win the return fixture 4-2 at Portman Road two days later.
Boxing Day 1991: Oldham 3 Man Utd 6
United ran riot at Boundary Park on the back of the lightning-fast breakaway pace of Andrei Kanchelskis and Ryan Giggs.
To add insult to injury for the hosts, two of United’s goals were scored by former Latics hero Denis Irwin, who emphasised the point with his jubilant celebrations.
Boxing Day 1992: Sheff Wed 3 Man Utd 3
The Owls appeared to be cruising to a famous win at Hillsborough as goals from David Hirst, Mark Bright and John Sheridan gave them a 3-0 lead.
But United, desperate to keep pace with shock Premier League leaders Norwich, roared back with a Brian McClair brace and one from Eric Cantona to steal a point.
Boxing Day 2007: Chelsea 4 Aston Villa 4
An incredible Stamford Bridge duel saw eight goals and three sendings-off.
Shaun Maloney gave Villa a 2-0 lead before Zat Knight was dismissed, sparking a Chelsea fightback to grab a point in a 4-4 draw – but not before Ricardo Carvalho and Ashley Cole also saw red
Liverpool fended off three goals in five minutes from Arsenal to salvage a 3-3 Premier League draw at the Emirates Stadium.
Roberto Firmino’s strike looped home off Petr Cech’s attempted save, to ensure the Reds left north London with a share of the spoils despite having earlier squandered a 2-0 lead.
Alexis Sanchez, Granit Xhaka and Mesut Ozil found the net in a stunning second-half triple salvo, to put Arsenal 3-2 ahead having trailed to goals from Philippe Coutinho and Mohamed Salah.
Here are our player ratings…
PETR CECH: A poor effort from the usually-reliable Arsenal stopper: 5/10
HECTOR BELLERIN: Always driving forward with pace, but not effective enough: 5
LAURENT KOSCIELNY (CAPT): Woefully out of sorts at the back when Liverpool threatened: 4
NACHO MONREAL: Another to struggle when Liverpool squeezed forward: 4
AINSLEY MAITLAND-NILES: Exposed out of position and up against the in-form Mohamed Salah: 4
JACK WILSHERE: A quiet night from one of the men expected to drive the pace: 5
GRANIT XHAKA: His goal a complete fluke that failed to mask a poor effort: 4
ALEX IWOBI: Struggled to find an influence on proceedings: 5
MESUT OZIL: Fine finish for the equaliser, solid enough overall despite poor first-half: 6
ALEXIS SANCHEZ: Grafted all night and took his goal well: 7
ALEXANDRE LACAZETTE: Played the key role in Ozil’s goal, and kept battling away: 7
THEO WALCOTT (for Sanchez, 88): Precious little time for an impact: 6
SHKODRAN MUSTAFI (for Monreal, 46): Another to struggle when Liverpool raced forward: 5
DANNY WELBECK (for Iwobi, 77): Could not make anything happen in a short cameo: 5
SIMON MIGNOLET: Awful effort all-round, mistakes galore and clearly at fault for Xhaka’s goal: 4
JOE GOMEZ: Quiet night overall even when Liverpool pushed forward: 5
DEJAN LOVREN: Failed to take charge when the pressure came on: 6
RAGNAR KLAVAN: Average effort overall as Liverpool buckled in the second-half onslaught: 5
ANDREW ROBERTSON: Unable to influence proceedings markedly at either end: 6
JORDAN HENDERSON (CAPT): Early injury robbed him of any lasting impact: 5
EMRE CAN: Put himself about and fought to provide continuity for the Reds: 6
MOHAMED SALAH: A constant threat and another fine showing from the in-form Egypt star: 8
PHILIPPE COUTINHO: Took his goal well and had an effective night: 7
SADIO MANE: Quiet by his lofty standards but still threatened with sheer pace: 7
ROBERTO FIRMINO: Put Arsenal under consistent pressure for much of the night: 7
GEORGINIO WIJNALDUM (for Mane, 79): Added drive with fresh legs: 7
JAMES MILNER (for Henderson, 11): Solid shift from the ultra professional: 7
ALEX OXLADE-CHAMBERLAIN (for Coutinho, 83): Unable to make a high-profile cameo against his former club: 6