Neckarelz, who switched the tie to Hoffenheim and were rewarded with a bumper 30,000 crowd, held the 14-times Cup winners for 50 minutes, although they were helped by Bayern's dreadful first-half finishing.
Gomez, who moved from VfB Stuttgart in May, broke the deadlock in the 51st minute with a shot which went in off both posts from Bastian Schweinsteiger's pass. His second came from a penalty awarded for handball six minutes later.
Despite having Christian Fickert sent off for a second bookable offence, the part-timers pulled one back through Heiko Throm with 10 minutes left before Hamit Altintop restored Bayern's two-goal lead and sent them into the second round.
The first round of the cup is a potential minefield for Bundesliga clubs as the seeding system pits them away to lower league teams in one-leg ties.
Hanover 96 became the second top flight side to go out when they lost 3-1 at fourth division Eintracht Trier.
Jan Rosenthal gave Hanover a halftime lead but Martin Wagner, Josef Cinar and Sierra Leoneon striker Sahr Senesie scored in the last half hour to give Trier a stunning win. Mainz were the other victims, losing 2-1 at fourth division VfL Lubeck on Friday.
Titleholders Werder Bremen had no such problems as they trounced second division Union Berlin 5-0 away with two goals apiece from Ivorian Boubacar Sanogo and Bolivian Marcelo Martins and one from Brazilian Naldo.
Martins was making his debut after joining on loan from Shakhtar Donetsk.
'Manchester United look confused about what they're being asked to do - he's certainly not getting the best out of them., I wouldn't bet a single penny on them getting to the next round' Alan Shearer issues damning verdict of Europa League hopes
'Pardew showed me a video of Shearer’s goals. Control, bam! Control, turn, bam! I was stunned, then said, "Give me that shirt, I always take number nine"': Newcastle cult hero recalls making confident move upon signing for Magpies 13 years ago
