Webber celebrated by taking his helmet off to enjoy the crowd's cheers on the slowing-down lap.
Vettel was in control of the race from the moment he re-took the lead at the start of the second lap from Mercedes driver Nico Rosberg, who had passed him off the line.
But a mix-up at his final pit stop on lap 47 cut his 12-second lead over Webber to just six and put Alonso right on the Australian's tail.
Red Bull had called Vettel in late fearing they might be caught out by a safety car following a collision between Mercedes driver Lewis Hamilton and Williams's Valtteri Bottas.
Hamilton hits Bottas in Brazil
The safety car never came, but the late decision meant Vettel's tyres were not ready when he arrived at his pit box and he sat there for several seconds while they were retrieved from the garage.
By the time they had been fitted, Webber was already stacked behind him and he only just got out in front of Alonso.
Alonso set a series of fastest laps straight after rejoining but Webber held him off and then began to edge away as both closed on Vettel.
For five laps after the stops, Webber inched closer to Vettel but when his lead had been cut to just 5.1 seconds, the German began to move clear again and the race was settled.
A sprinkling of rain in the closing laps was not enough to cause the drivers to have to switch to wet-weather tyres and Vettel cruised to victory.
Although it equalled Ascari in terms of number of consecutive wins, it was a new record for a single season, the Italian having achieved his victories spread over the 1952 and '53 seasons.
"I am actually quite sad the season is coming to an end," Vettel said. "The car has been phenomenal and has been getting better in every race.
"Thanks to Mark [Webber], we didn't have the best of relationships but we have always had a lot of respect for each other and we have been a very successful pairing for the team."
Hamilton had looked set for fourth place until his collision with Bottas.
The two touched on the entry to Turn Four as Hamilton, on the inside, edged slightly towards the Williams in the braking zone.
There appeared to be some room to the outside for Bottas to move into, but Hamilton was adjudged by the stewards to have crowded his rival.
The resultant drive-through penalty dropped him to 11th place but he fought back up to ninth at the finish.
Formula 1: The best moments of 2013
Jenson Button benefited by taking fourth place from 14th on the grid, the best finish of a dismal season for McLaren, with team-mate Sergio Perez sixth behind Rosberg.
Ferrari's Felipe Massa took seventh, despite a drive-through penalty for cutting the white line marking the pit entry, ahead of Sauber's Nico Hulkenberg.
Toro Rosso's Daniel Ricciardo, who is replacing Webber at Red Bull next season, took the final point in 10th place, ahead of Force India's Paul Di Resta, whose position in F1 next season is in doubt.
Vettel started from pole position but did not quite lead every lap - he made a slow start and as he defended from the fast-starting Alonso in third, Vettel was passed by Rosberg into the first corner.
But Vettel tracked his fellow German around the first lap, blasted past into the first corner at the start of lap two and never looked back.
Behind him, Alonso passed Rosberg for second into Descido do Lago at the end of the back straight on lap four, and by the time Webber has passed the German on lap six and then Alonso on 13, both into the first corner, Vettel was 10 seconds in front. Vettel controlled the race from there.
Webber lost second place again to Alonso after a delay fitting his left rear wheel at his first pit stop on lap 23, but grabbed the place back by re-passing Alonso into Turn One three laps later.
Alonso added: "It was a difficult race for everyone. It seemed like it was going to rain at any time but it didn't and it was a shame for us.
"But to finish on the podium means we finish the season on a high. Next year we look forwards."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.