The rampant Bavarians handsomely avenged their shock 1-0 first-leg last 16 defeat against the Swiss champions to go through 7-1 on aggregate and keep alive their dream of playing the final in their own Allianz Arena stadium.
It was the biggest winning margin in a Champions League knockout round match, Bayern's biggest win in the competition and Basel's heaviest European defeat.
That left the striker one goal short of Lionel Messi's record Champions League haul of five in Barcelona's match against Bayer Leverkusen last week.
It was Gomez's second hat-trick in successive games, after his three goals in the 7-1 win over Hoffenheim in the Bundesliga on Saturday, and took his Champions League tally to 10 goals.
"We were aggressive, strong and very clever against a team who can play much better than they did today," said Bayern coach Jupp Heynckes, whose side could still win a treble this season.
Basel's clearly shell-shocked coach Heiko Vogel, who suffered only his second defeat since taking over in November, added: "I'm not disappointed that we lost, but I'm enormously disappointed at the way we lost.
"We have achieved a lot, the team has achieved lot and we shouldn't forget this, I don't want our campaign to be reduced to this result."
Basel were unbeaten in their previous 17 matches but playing at a packed Allianz Arena was a far cry from games such as Saturday's Swiss League win over Grasshoppers in front of 7,000 people.
Muller's pass was defelcted through the Basel rearguard, leaving Robben with only goalkeeper Yann Sommer to beat which he did emphatically.
With Robben and Ribery allowed far too much space down the flanks, Sommer kept Basel alive as he twice denied Gomez, followed by a stunning reflex save to turn away Muller's header at point-blank range.
