Vettel's eighth victory of the year gives the Red Bull driver an opportunity to clinch his second Formula One title in the next race in Singapore later this month.
"It has been an incredible year so far, having progressed so much as a team," Vettel said. "We've got ourselves in a very good position."
Vettel clocked one hour 20 minutes 46.172 seconds during the 53-lap race, with McLaren's Jenson Button crossing 9.590 seconds behind and Alonso 16.909 back. It was the 18th victory of Vettel's career and second at Monza.
The key points in the race came early on, with Alonso making a risky move to take the lead on the opening straight after starting in fourth position on the grid.
Alonso put one tire of his Ferrari on the grass to get around McLaren's Lewis Hamilton and then beat Vettel to the first chicane.
Vettel wasn't about to stay second, however, and went around Alonso on the outside of the sweeping Curva Grande in the fifth lap, cruising to victory from there.
Button eventually passed Alonso to finish second and Alonso was third at the checkered flag with Hamilton fourth.
"We didn't have good starts the last few races, but we knew that here in Monza that if the car is in a good position there is enough space in the run to the first corner to overtake people," Alonso said. "But obviously we were not competitive enough to fight for victory, so we fought for the podium. It was a fantastic race."
Hamilton finished fourth, five-time Monza champion Michael Schumacher of Mercedes was fifth and Ferrari's Felipe Massa crossed sixth.
Also of note, Bruno Senna of Renault — whose uncle was the late Brazilian F1 great Ayrton Senna — finished ninth for his first points in F1.
Vettel became the youngest winner in F1 history here in 2008 at 21 with Toro Rosso but his fourth-place finish last year was Red Bull's best result in six attempts at the track — until now.
The memories of the 2008 win made Vettel grow teary-eyed during the victory ceremony, as thousands of red-clad Ferrari fans rushed below.
"It was very emotional — seeing all those people down there cheering and running," Vettel said. "It means a lot to me and when I crossed the line I remembered every single moment. ... The last two years we had a great car — but not here.
"This (race) is the best podium — the only thing that could be better would be wearing a red suit," the German added, revealing a desire to race for Ferrari someday.
Vettel leads the drivers' standings by 112 points, with teammate Mark Webber — who crashed out — tied for third with Button, 117 points back.
"It's over when it's over — not before. Last year is proof that you don't know until the last lap of the last race," Vettel said, referring to how he overtook Alonso for last year's title in the final race of the season.
Vettel's eight wins mean only Schumacher and Nigel Mansell have won more in a season. Mansell won nine in 1992 and Schumacher registered three seasons with nine wins, one year with 11 and established the record of 13 victories in 2004.
With average speeds of 155 miles per hour and top speeds of 211 m.p.h., Monza is the fastest circuit on the calendar, and drivers got optimal use out of their drag reduction systems (DRS) — adjustable rear wings — and the KERS power boosts, creating a race filled with overtaking.
But, as is so often the case, the race began with chaos.
Further back on the starting grid, Vitantonio Liuzzi slid his HRT across the grass and into Vitaly Petrov's Renault and Nico Rosberg's Mercedes at the first chicane, bringing out the safety car and ending the race for all three drivers.
Red Bull's Mark Webber then attempted to pass Ferrari's Felipe Massa on the sixth lap and made contact. Massa emerged seemingly unscathed but Webber lost his front nose and crashed later in the lap to end his race.
Hamilton and five-time Monza champion Michael Schumacher traded places twice in a tight battle for third, with Hamilton eventually remaining in front, but the British driver then made an error that allowed Button to surge past, as Schumacher dropped back after his first pit stop.
Button passed Alonso with 16 laps to go and the Spaniard held off a pressing Hamilton on the final lap.
Alonso got Ferrari's only win this year at the British GP in July, and with the home team struggling attendance in Monza appeared slightly down from past years.
The Ferrari fans who came to the track north of Milan posted banners offering encouragement, with many saying "We'll always be with you."
Alonso bowed to the crowd during the podium celebration and received a warm applause from the fans — more affectionately known as "tifosi."
The race was held in clear and warm conditions, with the track temperature soaring to 40 C.
Daniel Ricciardo of HRC, Jerome D'Ambrosio of Virgin and Adrian Sutil of Force India had their races cut short, each with apparent technical problems.