Pole-sitter Raikkonen looked set to win his first race in more than five years - and Ferrari's first in their homeland since 2010 - but Hamilton executed a brilliant pass on the Finn with just eight laps remaining.
Ferrari had the quickest cars at Monza and Vettel and his teammate Kimi Raikkonen had locked out the front row and were expected to win.
Lewis Hamilton said winning the Italian Grand Prix and extending his championship lead to 30 points was beyond his hopes going into the race. "Then I was the one who spun around, which was ironic".
"When you're looking the wrong way, things do not look too peachy but I recovered and got a lot of points", he added.
Bottas said: "I was really trying to do everything I can to get to the podium - as a team, we take this result".
Hamilton said: 'I was a bit surprised Seb chose the inside and not the outside of Kimi.
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Atletico Madrid, AS Monaco and Borussia Dortmund were drawn together in an evenly-balanced Group A completed by Club Bruges. However, Croatia captain and midfielder Modric said he could not be happier with the Champions League holders.
The Dutchman was handed a five-second time penalty by which time Hamilton was leading, having passed Raikkonen on the outside at the Rettifilo.
"I stuck it down the outside and had to make sure I was far enough alongside - I had an experience where I was on the inside back in 2012 or something like that and it didn't come off very well, but I thought there was enough space between us".
"[It] didn't help to be behind the other cars for some laps, and it wasn't really like you can save an terrible lot of tyres, you are not in a position to start saving and taking it easy - well you can, but you let him [Hamilton] past", he said.
"It was unfortunate for us to get spun around and have a lot of damage but it could have been also him that spins around and us carrying on".
The race re-started after three laps with Hamilton attacking Raikkonen for the lead and the stewards concluding that the battle for the lead had been "a racing incident". I am guessing he locked up, we touched, it was a brief moment, damaged the vehicle a little bit but I could continue.
"I'm not too anxious", he told reporters after finishing fourth in an Italian Grand Prix won by his Mercedes rival. "Lewis left him enough room".
"To know that I am up there with Michael is always a real privilege". And then you get back up and you start from the back and you have to come through. "It is very hard for people watching who are not in it to feel what we feel".