Hannover's Ivorian Didier Ya Konan slotted home from close range in the 57th minute and Senegalese Mame Biram Diouf headed in his fifth league goal this season in the 77th before Gladbach cut the deficit a minute later through Havard Nordtveid.
The win put Europa League quarter-finalists Hanover, who trail Atletico Madrid 2-1 after the away leg, firmly back in contention for a spot in the competition next season.
They are on 41 points in fifth place while Gladbach, after their third defeat in the last five games, are stuck in fourth on 51 with six games left to play.
"We made far too many mistakes," said Gladbach coach Lucien Favre. "We need to rediscover those 15 percent that are currently missing in our game."
The top three teams automatically book a spot in the Champions League group stage for next season.