He was more emotional than usual, shaking his head vigorously in celebration and letting out his trademark high-pitched whoop as he celebrated his third win in the past four races, then pumping his fist several times as he walked up the podium steps.
Massa finished fourth ahead of German drivers Nico Hulkenberg — a season's best for Sauber — and Nico Rosberg, who was sixth on a poor day for Mercedes.
There was light rainfall before the race, but the sunshine returned as Vettel started from pole for the 40th in his career and first since the Canadian GP in June.
Roared on by thousands of flag-waving tifosi, Ferrari had a point to prove and Alonso brilliantly overtook Webber on the outside and then zoomed ahead of Massa.
After 10 of 53 laps, Vettel led Alonso by about 5 seconds, but engineers told him that his front right tire had taken some damage on turn 1 and Alonso began to make up time.
Vettel pitted on lap 24, switching from medium to hard tires, followed shortly by Webber and Massa, but Alonso stayed out until lap 28 as leading drivers chose a one-stop strategy.
In his last European race before he retires from F1 at the end of the season, Webber got his first podium at Monza.
"Good battle with Fernando," the Australian said. "Last time in Monza, so thank you very much."
Briton Paul di Resta's race lasted a few seconds after he sustained damage to the front left tire of his Force India car. He appeared to hit the back of Romain Grosjean's Lotus and stewards were investigating the incident.
Hamilton started from 12th after his worst qualifying performance this campaign and finished ninth.
"That's it for the Championship now," said Hamilton, who won the F1 title in 2008.
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