It took some time for Hanover to find their way back into the game but they did so with determination when Didier Ya Konan cut the deficit with a fine volley in the 37th minute when defenders Dante and Daniel van Buyten were out of position.
Mueller, however, made sure of Bayern's spot in the next round when he connected with a Xherdan Shaqiri free kick to grab his second goal in the 64th.
Substitute Franck Ribery added another when keeper Ron-robert Zieler failed to hold a powerful Mueller drive 12 minutes from time.
Benedikt Hoewedes rose high to head the visitors back into the lead on the hour after Jefferson Farfan's 35th-minute lead was cancelled out by Hanno Behrens' header a minute later. Max Meyer sealed the win in the 86th.
Second tier Kaiserslautern came from a goal down to dump out Bundesliga Hertha Berlin 3-1.
Unmarked Peter Niemeyer controlled and drilled in to put Hertha ahead in the 25th minute but the hosts bounced back in the second half, scoring three times through Mohamadou Idrissou, Karim Matmour and Canadian Olivier Occean.
Last season's finalists VfB Stuttgart also bowed out with a 2-1 defeat at Freiburg.
Eintracht Frankfurt comfortably beat second division VfL Bochum 2-0 with Japanese Takashi Inui scoring one goal and setting up another in the space of two minutes.
Borussia Dortmund left it late against 1860 Munich on Tuesday, scoring twice in extra time to advance.