Hamilton is 17 points clear of his rival ahead of the sport’s final round before the summer break in Hungary on Sunday.
To date, the Englishman, 33, holds the mental edge over Vettel following the Ferrari driver’s fourth, and biggest mistake of the year, when he crashed out from the lead in Germany last Sunday.
Hamilton’s remarkable victory in Vettel’s back yard moved him to the summit of the standings to mark the fifth change of championship leader this season.
For the first time in the hybrid era, Ferrari, who are sporting a black stripe on their cars this weekend in memory of chairman Sergio Marchionne who died suddenly on Wednesday, boasts the strongest package.
That was again on show in practice here as Vettel topped the time sheets with Hamilton only a distant fifth.
By his own admission, the Briton has not been at his very best this year, but it is his lack of mistakes, and flashes of once-in-a-generation brilliance, that ensure he remains the favourite to clinch a fifth world crown.
“The psychological game in sport is the hardest thing,” Hamilton said. “You don’t see me away from the track. I wake up with insecurities.
“The most demanding thing is keeping your mind in the game from the first race in March right through to the November, and arriving at every race 100 per cent.
“I haven’t hit the nail on the head at every weekend this year, but the pressures are huge, and the demand on myself and Sebastian is higher than ever.
“It has swung more in Ferrari’s direction so I am having to over-deliver. The pressure to extract every ounce is greater than ever if I want to be number one at the end.
“That is not something I am fazed by, but something that excites me. I have always felt that I have been at my best under pressure. I welcome it.”
Vettel, but for his mistakes, should be leading this championship. He dropped from second to fourth after running off the road in Azerbaijan.
He collided with Valtteri Bottas in France, and was demoted five places on the grid at the ensuing race in Austria after blocking another driver in qualifying.
Last weekend, with Hamilton catching him in the slippery conditions, Vettel threw away a certain 25 points.
“It is closer than last year, so the smallest mistakes are even more costly,” Hamilton added. “But I take pride in being a perfectionist and not making mistakes.
“Nobody is perfect, but I work to position myself mentally and physically so I am the last to crack.”
There have, however, been signs of the intensity getting to Hamilton. He wildly accused Ferrari of playing dirty at Silverstone (Kimi Raikkonen collided with him on the first lap), while his reaction after his car broke down in qualifying last weekend was as if he had lost the title.
In a deleted Instagram post, he then said he did not get the credit he deserved from Sky Sports for his comeback fight from 14th to first last Sunday.
“Positive headlines don’t sell newspapers, and don’t generate clicks,” Mercedes boss Toto Wolff said. “In the here and now the great achievements are not recognised how they should be.
“In five or 10 years, we will look back and say we were part of an amazing driver’s journey. He wears his heart on his sleeve, and polarises opinion, but for me that is fine.”
Vettel should have moved 21 points ahead of Hamilton at the German Grand Prix, but will instead head to Hungary 17 points behind the Briton after his dramatic retirement at Hockenheim on Sunday.
The four-time champion crashed off in the slippery conditions, and, to make matters worse, his rival Hamilton sealed a famous victory from 14th on the grid.
“What an unbelievably dark day for Sebastian,” Rosberg said. “It was surely one of the darkest moments of his whole career.
“It was his home race, he had the chance in his hands to win and take a big lead in the championship, but he chucked it away in the wall.”
Vettel has been found wanting in the heat of his championship battle with Hamilton. His potentially title-deciding mistake on Sunday was his seventh high-profile error in a little more than 12 months. During the same period, Hamilton has not made a significant faux pas, and it is proving the difference between the two quadruple world champions.
Ferrari team principal Maurizio Arrivabene did not mention Vettel by name in Ferrari’s post-race press release.
The Italian team headed into Sunday’s race rocked by the news that their 66-year-old chairman Sergio Marchionne had been stood down from his role through illness.
“In what was a particularly fraught weekend for Ferrari, it would have been important for us to bring home the win,” Arrivabene said. Perhaps tellingly, he added: “Our car had shown it was up to the job.”
Vettel is paid £38million a year by Ferrari, but his mistakes are costing both himself and the team dearly.
Despite his demise, it still required all of Hamilton’s brilliance to claim the 66th win of his career.
“Never write off Lewis because he always comes back, and that is a fact,” added Rosberg, whose relationship with his former team-mate remains frosty.
“He showed it again on Sunday by taking home the win.”
Hamilton’s championship defence had been derailed by a mechanical failure in Austria before he was taken off by Kimi Raikkonen on the opening lap of the British Grand Prix a fortnight ago. His car then broke down in qualifying on Saturday.
“It’s too early in the season to ever really feel like it’s slipping away but it never feels good when you face adversity,” Hamilton said.
“It felt at one point, that, jeez, there was a steep hill for us, but the longer you endure it, the stronger you grow.
“I guess it is a crucial point in the season. It has been up and down, back and forth, but the mistake from Sebastian has been a benefit for us and Mercedes.”
Hamilton started only 14th after a mechanical failure in qualifying, but took the chequered flag at Hockenheim following a dramatic conclusion sparked by a rain shower, and Vettel’s jaw-dropping crash.
Both Hamilton’s Mercedes team-mate Valtteri Bottas and Ferrari’s Kimi Raikkonen pitted for new tyres after the safety car was deployed following Vettel’s costly accident.
Hamilton stayed on track to assume the lead, and when the race was back under way with 10 laps to run, the Englishman held off team-mate Bottas to seal a famous victory and afterwards thanked God for the win.
He said: “It is very difficult (to win) from that position and highly unlikely but you have always got to believe. I did a long prayer before the race and I wanted to stay collected and calm.
“I am so grateful. I kept pushing, kept believing and it happens. A big thanks to God.
“I hadn’t thought about the championship. It was so tough out there. Conditions were perfect because when it rained, I knew that I would be in a good position.”
Incredibly, the British driver is now 17 points ahead of Vettel after striking a devastating championship blow on his rival’s home turf.
Pole-sitter Vettel had been in control of his home race, and looked destined to increase his title advantage over Hamilton before a late rain shower ensured a spectacular end.
The Ferrari driver lost control of his car in the slippery stadium section towards the end of the lap, and slid across the gravel and into the tyre barriers.
Vettel was furious with his mistake, bashing both hands on the steering wheel. “F*** sake, f*** sake,” he yelled over the radio. “Sorry, guys. S***.”
The four-time champion got out of his Ferrari cockpit, and kicked the gravel in utter frustration.
To make matters even worse for Vettel, his potentially championship-deciding error, gifted the lead to Hamilton, who was running in fourth place.
Bottas and Raikkonen, both on older rubber, had to pit for new tyres, but Hamilton – despite being told to stop too, only before changing his mind at the last moment – stayed out on track.
The race resumed with just 10 laps to run, and Mercedes held their breath as Hamilton and Bottas went wheel-to-wheel for the victory.
But, at the team’s home race, Mercedes called off the fight: “Valtteri, this is James (Vowles – chief strategist). Please hold position.”
The Finnish driver duly obliged, and, despite the threat of a second rain shower, Hamilton crossed the line 4.5 seconds clear of his team-mate to move clear of Vettel in their battle for the championship.
Raikkonen completed the podium places, while Max Verstappen finished fourth ahead of the Renault of Nico Hulkenberg.
“Miracles do happen,” Hamilton’s race engineer Pete Bonnington said to the Brit after he took the win.
“What an amazing job from you guys,” Hamilton replied. “Thank you so much. Love conquers all.”