By the end he had lapped Mercedes' Hamilton, who was ninth and heading for his fourth title, and closing in on Vettel who finished fourth.
"We knew that it would be difficult for us here, running at a high downforce circuit, so the result is okay with that in mind; it feels like we have taken a step forward in these conditions compared to recent races".
Meanwhile Vettel needed an absolute minimum of a top two finish to extend the race.
The trio rejoined in the same positions, with Vettel and Hamilton also pitting for fresh rubber and taking ultrasofts and supersofts respectively with a little over half of the race to go.
Meanwhile Vettel fell to 19th place after colliding with Hamilton and managed to pull himself back up to fourth place.
Verstappen triumphed from Valtieri Bottas in the other Mercedes with Ferrari's Kimi Raikkonen third.
Stewart, the man who Hamilton replaced as Britain's most successful driver in terms of world titles, believes his compatriot is a worthy successor.
Federer's 95th ATP tournament win - the eighth in his hometown Basel event - lifted him out of a tie with Ivan Lendl, trailing only Jimmy Connors, who has 109.
"This was a championship I was hoping for, fighting another team", Hamilton said.More news: North Korea to Test Hydrogen Bomb Over Pacific
Verstappen, with nothing to lose and everything to gain from his front row position, seized the lead with an aggressive move through the opening right-left-right corners and the Red Bull bumping wheels with Vettel as he went through. "It is a long way down to turn one so we should have some fun tomorrow". That is what is important and in my heart. Hamilton, who had just hoped to limit his damage that day, seized on the chance and won. On Sunday, it was Vettel's bump with Hamilton that damaged both cars and forced both to quickly pit for repairs. After battling Rosberg the previous three seasons in a rivalry that split a childhood friendship and created tension within Mercedes' garage, racing against Ferrari was fun, according to Hamilton.
McLaren elected not to run in Q2 on account of their grid penalties which will see both Fernando Alonso and Stoffel Vandoorne demoted to the back of the grid.
Caroline Wozniacki won the biggest title of her career when she beat Venus Williams 6-4, 6-4 in the WTA Finals in Singapore.
The Briton joins Vettel and Alain Prost on four world titles, with only Michael Schumacher and Juan Manuel Fangio ahead on the all-time list with seven and five respectively.
The incident robbed the crowd of the prospect of a real duel between the two contenders, who will both be four-times champions when next season starts, but they still provided thrills as they fought back.
So how realistic is it that Hamilton could overtake Schumacher's wins record? "And his career is not done yet". Big thank you to my family, my team, Mercedes.
Team boss Christian Horner said he could tell from the look in his eyes the moment Verstappen arrived in the paddock in Mexico that he wanted to win more than anyone.
"You need to let it run wild and free and discover yourself".
Red Bull were considered a distant third in the 'top three' teams this year.