Rain early in the race allowed Vettel to establish a solid lead over Kovalainen, which he extended as the track dried.
Vettel's record was broken by Max Verstappen aged 18 years and 228 days at the 2016 Spanish Grand Prix.
Massa scored one point more than Hamilton, narrowing the McLaren driver's lead in the Championship once more with four races remaining.
However, Kovalainen's second-placed finish put McLaren closer to catching Ferrari in the Constructors' Championship.
[2] After the previous race, Hamilton was penalised for cutting a chicane and gaining an advantage over Räikkönen in the closing laps.
[5] Reflecting on the controversy, Norbert Haug, Vice President of Mercedes-Benz Motorsport, said: Our disappointment was big, when the stewards took away victory from him [Hamilton] and the team.
When we went to the airport last Sunday evening, Lewis said to me - preferably we now want to win all remaining races, don't we?
"[7] The close nature of the Championship meant that Ferrari's home race had the potential to be a turning point in the season.
[8] Testing at the Monza circuit in early September had indicated that the performance gap between McLaren and Ferrari had narrowed.
[8] Ferrari confirmed on September 12 that Räikkönen would continue to drive for the team until at least 2010, ending media speculation that he could be about to retire.
[12][13][14] Adrian Sutil was quickest with a time of 1:32.842 in the first session, more than half a second quicker than the next fastest drivers Rubens Barrichello and Giancarlo Fisichella.
[12] The track dried out in the second session; Räikkönen adapted best to the changed conditions with a lap of 1:23.861, significantly quicker than his morning time.
[13] Glock was fastest in the final session with a time of 1:35.464, followed by Sebastian Vettel, Rosberg, Jarno Trulli, Kazuki Nakajima and Heidfeld.
[19] Both Räikkönen and Hamilton blamed a late entry into the second session for their poor qualifying times, which saw them eliminated in 14th and 15th positions, respectively.
[22] Jenson Button and Kazuki Nakajima started the race from the pit-lane, their teams choosing to make adjustments from qualifying to the set-ups of their cars.
The race began at 14:00 local time (UTC+1) behind the safety car because of the rain, meaning that there would be a rolling start.
[27] Vettel immediately opened a lead of two seconds over Kovalainen; the McLaren driver experienced visibility problems as a result of the spray off the back of the Toro Rosso.
Emerging ahead of the Ferrari, he gave the place back, returning the advantage and avoiding a penalty.
As a result of the contact, the Force India's front wing broke loose and lodged under the car.
[22][29] Hamilton passed Heidfeld, Glock, Kubica and Alonso in separate manoeuvres, to sit in seventh by lap 19.
[27] Räikkönen continued to move through the field; he passed Coulthard and Piquet in separate manoeuvres to take ninth position.
[27] Vettel crossed the finish line on lap 53 by 12.5 seconds over Kovalainen, becoming the youngest driver in Formula One history to win a Grand Prix.
Vettel acknowledged the support of his team, saying the set-up of the car perfectly suited the way the race developed.
[36]Hamilton was satisfied with his performance in the race, saying "today was all about damage control: I came away with some points and kept my lead in the world Drivers' Championship.
"[28] Massa said that he had hoped to do better, but added "I've managed to make up a point on my nearest rival so this is absolutely not a negative result.
[2] Kovalainen's second-placed finish closed the gap in the Constructors' Championship between Ferrari and McLaren from twelve to five points.
[36] The Polish driver said that racing in Italy was particularly special for him, having lived in the country for five years: "That is why we divided the helmet into two pieces.
"[36] Alonso, who used a similar strategy, said "the result today is very good because Monza was supposed to be the race that we were looking forward to the least.