Mueller’s golden run continued at the Allianz Arena to leave him with seven goals and nine assists in the 14 games since Hansi Flick replaced Niko Kovac as coach last November.
Not to be outdone, Mueller’s strike partner Robert Lewandowski claimed two goals in the third-round tie.
However, Hoffenheim’s Israeli forward Munas Dabbur caused Bayern a few nervous moments with two late goals as the visitors threatened a fightback from 4-1 down.
“In the second half, we gave too much ball away and let them come back into the game,” admitted Bayern winger Serge Gnabry.
“We have to look at that, however we had a lot of fun going forward in attack.”
It was Mueller who had calmed Bayern’s early nerves, forcing an own goal from Benjamin Huebner to equalise after Bayern defender Jerome Boateng had turned the ball into his own net.
Mueller then gave Bayern a 2-1 lead before Lewandowski netted either side of half-time to extend their lead before Hoffenheim’s late fightback.
Bayern are the biggest name into the last eight after RB Leipzig and Borussia Dortmund both crashed out on Tuesday to Eintracht Frankfurt and Werder Bremen respectively.
After Lewandowski had an early goal disallowed for offside, Hoffenheim threatened an upset when Boateng turned a shot by Hoffenheim winger Ihlas Bebou into his own net on eight minutes.
However, the lead lasted just five minutes.
Canadian winger Alphonso Davies crossed to Mueller in the middle, but the ball clipped the knee of Huebner and looped into the net.
Bayern were 2-1 up on 20 minutes when David Alaba’s cross found Mueller and the Germany star fired home from close range.
Lewandowski then pounced when Hoffenheim goalkeeper Philipp Pentke failed to claim a high ball under pressure from Mueller.
The Poland striker was on hand to head the stray ball into an empty net on 36 minutes and appeared to have settled the matter with a well-taken fourth goal after the break.
Lewandowski has scored in all four of Bayern’s matches so far in 2020 and boasts a season tally of 35 goals in 29 games in all competitions.
Union Berlin are also through after seeing off fourth-tier minnows Verl with a 1-0 win thanks to a late goal by midfielder Robert Andrich.
A goal by Argentinian striker Lucas Alario helped Bayer Leverkusen to a 2-1 win at second-division side VfB Stuttgart.