Vettel locked his tires on the first turn as the Ferrari of Felipe Massa pressured him from behind, but after that the German consolidated his advantage to beat Alonso by 5.4 seconds and Red Bull teammate Mark Webber by 6.3.
"This is the best podium of season, I'm very proud," Vettel said. "We had problems with the gearboxes at the end, but I was OK because I had a good cushion and it was a fantastic win."
Monza is the home race for Ferrari fans and there were loud jeers when Vettel was on the podium.
"You can hear the difference when you don't win here in a red suit, but it means you have done well and beaten the red men (of Ferrari)," Vettel said. "The more booing we get the better we've done today. I don't blame the people to be honest, it's in their genes. They love Ferrari."
With seven races remaining, it looks increasingly likely that Vettel will clinch a fourth straight Formula One title — unless he has an alarming drop in form or Ferrari finds a way to get faster.
"It's a very big gap and we don't have enough races and we don't have the speed now to win consecutive races," Alonso said. "We need to be lucky and we need to have some DNF from Sebastian."
It was Vettel's third career win at Monza — his first was with Toro Rosso in 2008 and then Red Bull in 2011 — his sixth of the season and 32nd overall, moving him even with Alonso's win total.
He was more emotional than usual — pumping his fist several times as he walked up the steps to the podium — and with good reason. He now leads two-time former champion Alonso by 53 points and Lewis Hamilton by 81 with 175 available.
The last driver to win here from anywhere other than the pole was Rubens Barrichello from fifth spot on the grid in 2009, and Alonso never showed he was likely to do that starting from the same position.
"Second place is good, to have this podium ceremony which is the most spectacular podium of the year," Alonso told thousands of his fans. "Hopefully we'll come back next year and have first place."
Vettel shook his head vigorously in celebration and let out his trademark high-pitched whoop as he celebrated his third win in the past four races. Given his tough start, it was more impressive than his dominant performance at the Belgian GP two weeks ago.
"Very good boys, very good race," Vettel yelled over his car radio when he crossed the line. "Sorry for the lock up into the first corner, made the first stint a bit tricky."
Massa finished fourth ahead of German drivers Nico Hulkenberg — a season's best for Sauber — and Nico Rosberg, on a poor day for Mercedes.
There was light rainfall about one hour before the race, but the sunshine returned for the start.
Vettel started from pole for the fourth time this campaign and 40th in his career, but the first since the Canadian GP in June.
He struggled to hold on to the lead after the first turn as Massa boldly passed Hulkenberg and Webber to jump from fourth to second.
Roared on by thousands of flag-waving tifosi, Ferrari had a point to prove and Alonso brilliantly overtook Webber on the outside to move up to third 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 the first turn and Alonso began to shave some time off.
Vettel pitted on lap 24, switching to hard tires, followed shortly by Webber and Massa, but Alonso stayed out on his tires 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," Webber said. "Last time in Monza, so thank you very much."
Lewis Hamilton, starting from 12th after his worst qualifying performance since the Spanish GP in May of last year, finished ninth.
"It is a nightmare weekend for me," Hamilton said.
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