Leverkusen went ahead after two minutes through Gonzalo Castro but Huszti equalised from the spot in the 20th minute.
Mame Biram Diouf gave Hannover the lead 2-1 in the 57th minute before Stefan Kiessling levelled 60 seconds later, but Huszti converted another penalty in the 69th minute.
The result ended Leverkusen's four-match winning run and left them 11 points behind runaway leaders Bayern.
Two goals within five minutes were enough for Borussia Monchengladbach to win at home against Mainz and move closer to the European places.
Mike Hanke scored in the 58th minute and Juan Arango netted a memorable goal in the 63rd to decide the contest.
Hanke turned home Tony Jantschke's cross before Arango collected a clearance from outrushing Mainz keeper Heinz Muller and flighted the ball back over the keeper and in from 45 yards.
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