Gomez scored three times in 25 minutes to take his Champions League goal tally to 17 and become his country's all-time top scorer in the competition, surpassing Michael Ballack's 16.
"We played a terrific first half and that was a lot of fun," a beaming Gomez told reporters. "And I am the one in the middle of all this."
The Italians managed to pull two goals back through Federico Fernandez in a bad-tempered second half in which Napoli's Juan Zuniga and Bayern's Holger Badstuber were both sent off after receiving second yellow cards.
Bayern, who made it 10 wins from their last 11 Champions League matches, will be without Schweinsteiger for at least four weeks as he will need surgery on his collarbone.
The Bavarians top the group with 10 points with Manchester City, who beat Villarreal 3-0, second on seven with Napoli third on five. Villarreal have yet to win a point.
"We saw a game with two different halves. We had a very fluid first half, then the second half was different. But still I believe we have to win such games as well," Bayern coach Jupp Heynckes told reporters.
"It shows the team has a strong spirit and at the end we deserved to win tonight."
"Mario today was not only a great player because of the goals but his overall play was outstanding. It does not happen better than tonight," Heynckes said.
EXPLOSIVE START
The Bavarians, who have scored an eye-popping 18 goals in their last four league home games, needed 17 minutes to settle in on a bitterly cold evening, taking the lead when Gomez picked up a Schweinsteiger pass, wrong-footed defender Hugo Campagnaro and fired in a low drive.
It was Bayern's 250th goal in the Champions League, only the fourth team to reach this number after Real Madrid, Barcelona and Manchester United.
Germany international Gomez, who has also notched 12 league goals in 11 matches this season, then struck again six minutes later volleying in a lovely chip from Toni Kroos.
With Napoli desperately scrambling to figure out how to stop the Bayern onslaught, the hosts came close again, this time with Franck Ribery dribbling into the box only to see his shot from a tight angle palmed away by goalkeeper Morgan De Sanctis.
Gomez should have scored a third when he raced clear with only De Sanctis to beat but his flick went wide.
However, he did complete his hat-trick after 42 minutes, benefitting from confusion in the box to slot in from close range for his 13th goal in the last 10 Champions League games.
Fernandez cut the deficit for the Italians just before the break with a well-timed header as the hosts gradually stepped off the gas and lost their focus.
The Argentine struck again 10 minutes from time, heading in as the Bayern defence left him unmarked at the far post to trigger a nervous finish to the game.
Gomez almost got a fourth with a
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