The 21-year-old German driver defeated Heikki Kovalainen of McLaren by 12.5sec on the slick Monza circuit to give Toro Rosso its first win.
BMW-Sauber's Robert Kubica was third ahead of Fernando Alonso of Renault, while Nick Heidfeld of BMW-Sauber was fifth.
Briton Lewis Hamilton worked his way up from 15th on the grid to finish seventh, maintaining his lead in the overall drivers' standings.
Felipe Massa of Ferrari was sixth and now trails the Hamilton by just one point with four races left. Australia's Mark Webber finished eighth for the final point.
Neither Vettel nor his Toro Rosso team could contain their emotions as the young driver emerged on to the podium with fists pumping.
Vettel never lost his nerve during the wettest race at Monza for 27 years as he broke Alonso's mark of F1's youngest winner. Alonso, a two-time world champion, was 22 when he won the 2003 Hungarian Grand Prix.
Vettel will partner Webber at Red Bull next season as he replaces the retiring David Coulthard.
The safety car was brought out for the start, with the steady drizzle failing to let up and all cars using extreme wet tyres to navigate the damp track.
Vettel, who also was the youngest driver to start from pole, held his position over Kovalainen on the flying start thanks to a clear track and none of the poor visibility that the rest of the field had to deal with from the resulting spray.
Hamilton pitted later than many of his rivals and was trailing only Vettel by the time he stopped to refuel, while defending champion Kimi Raikkonen was stuck behind the traffic in 10th by his first stop.
Hamilton pushed Webber out down the straight in the closing stages with the cars touching wheels to force the Red Bull driver off the track.
Only Force India's Giancarlo Fisichella retired despite the slippery conditions.