"You've won it in style," an elated Red Bull team principal Christian Horner told Vettel over the car radio immediately after the race. "Brilliant stuff. You've joined the greats."
After taking the checkered flag, Vettel performed several donuts on the track before climbing from his car, acknowledging the crowd and kneeling with his head bowed to the track. He then climbed the pit lane fence to embrace members of his team.
"I'm overwhelmed, don't know what to say," Vettel said immediately after the race. "It is one of the best days of my life.
With Vettel's win, Red Bull also claimed its fourth straight constructors title, despite losing Mark Webber to mechanical problems while in second place.
Ferrari's Felipe Massa was fourth with teammate Fernando Alonso finishing a disappointing 11th.
"Today is the day to congratulate (Vettel) and for him to enjoy the championship," Alonso said, who is 115 points behind the Red Bull driver in second place in the standings with three races remaining.
Vettel has now won 10 races this season, including the last six.
A clearly emotional Vettel, taking occasional swigs from his celebratory magnum of champagne, thanked his team and his family in a post-race media conference. He spoke of his respect for the Indian people and rivals including Alonso, Lewis Hamilton and Webber.
"To win four titles, I don't know, it's just a big number you know?" Vettel said. "Fangio put the number of five titles and everybody appreciated him as the best driver in the world. Michael (Schumacher) came along ... it's incredible that one guy managed to score more championships (seven) than (Fangio) did."
Starting on the quick but fragile soft tires, Vettel lasted just one lap before switching to the medium compounds, emerging from the pit lane in 17th place.
Vettel wove his way back toward the front, gradually narrowing the gap and passing McLaren driver Sergio Perez on the 22nd lap to move into second behind Webber.
Webber eventually entered the pits for his first tire change to soft tires on the 29th lap and Vettel made his second and last stop three laps later to new medium tires, returning to the track just over eight seconds behind Webber.
When Webber switched back to medium tires two laps later, he effectively handed the race to Vettel.
Red Bull then ordered a stunned Webber to stop while in second place on the 42nd lap due to a problem with his car's alternator.
"Disappointing, but there's not much I can do," Webber said. "There was something wrong with the car, so we had to stop straight away."
With the Australian out, the race became a tussle for the lesser podium places, with pacesetting Rosberg passing Lotus driver Kimi Raikkonen for second on the closing laps before Grosjean capped a dogged race by defying the odds and reaching the podium after being eliminated in the first qualifying session Saturday.
"I would not have bet a penny on me being on the podium today," Grosjean said. "But the car came back. "
Vettel meanwhile was struggling to come to terms with the enormity of his achievement.
"It was a difficult emotion to cross the line and feel happy all of a sudden because you're in a rhythm, you know what you're doing, you have a routine," he said. "I think it takes time to understand what happened."
