Toni Kroos and Mario Gomez gave the hosts a two-goal cushion before Didier Konan's 74th-minute bicycle-kick sparked late drama on a rainy day in Munich.
"I have to give credit to the team because it is hard a couple of days after Wednesday's game [German Cup semi-final win on penalties over Borussia Monchengladbach] to perform like they did today," said Bayern coach Jupp Heynckes.
"We played clever football for much of the game and deserved the win."
Schalke 04 jumped into third place and an automatic Champions League qualification spot with a 2-0 win over last season's runners-up Bayer Leverkusen. Dutchman Klaas-Jan Huntelaar scored both goals to take his league tally to 22.
Borussia Monchengladbach dropped to fourth on 51 points, two behind Schalke, after conceding two goals in two minutes against Hoffenheim to lose 2-1 in their first home defeat in a year.
Second-placed Bayern, who left Bundesliga top scorer Gomez on the bench at the start, had a good chance in the 20th minute but Franck Ribery fired wide from 10 metres.
The Bavarians, up against Olympique Marseille next week in the first leg of their Champions League quarter-final, went ahead in the 36th minute through Kroos.
The Germany international chipped the ball over goalkeeper Ron-Robert Zieler after a quick exchange between Ribery and Arjen Robben.
Gomez, who came on after an hour, went close when his low drive from a tight angle rattled the post.
The Germany forward was then on target in the 68th minute, grabbing his 23rd league goal of the season after shaking off two defenders to curl the ball past Zieler with a fine left-footed effort.
SPECTACULAR GOAL
Substitute Konan spectacularly cut the deficit for Hanover with a bicycle-kick in the box in the 74th minute to trigger a nervous finale for the home team.
Mame Biram Diouf then went close to an equaliser when his header flew narrowly over.
The hosts were given another fright when Bayern defender Jerome Boateng miskicked, forcing his own keeper Manuel Neuer into a reaction save.
Bayern are chasing a trio of titles after qualifying for the German Cup final where they will face Dortmund.
Relegation-threatened Hertha Berlin, in 17th spot, grabbed their first away win since October when they beat Mainz 05 3-1 with two goals from Adrian Ramos
The capital club moved to within a point of the relegation play-off spot occupied by Hamburg SV.
Kaiserslautern's new coach Krasimir Balakov could do nothing to prevent a 2-0 defeat at Freiburg in their relegation derby that left them 17 league games without a win - a club record - and anchored in last place on 20 points.
Augsburg snatched a last-gasp 1-1 draw at sixth-placed Werder Bremen thanks to Paul Ve
