Werder Bremen were desperate to turn things around after just one victory in their last eight games before Sunday, but fell behind when Szabolcs Huszti put the visitors ahead from the penalty spot on 20 minutes.
Aaron Hunt cancelled out the advantage with a spot kick of his own five minutes later and Cedrik Makiadi then put Werder in front with a header.
Hanover levelled with a powerful shot from Japanese Hiroki Sakai four minutes before the break, but were undone in the closing stages.
A goalmouth scramble allowed crowd favourite Garcia to score an 85th-minute winner from a metre out that lifted Werder up to eighth spot on 15 points.
Hanover slipped to 11th on 13 following their third straight loss and their fifth league game without a win.
Augsburg notched their first league win since September to snap a five-game streak without a victory and beat Mainz 05 2-1 with Andre Hahn scoring both goals to lift them up to 13th place.
Bayern Munich remained in control of the Bundesliga with a narrow 2-1 win at Hoffenheim on Saturday to stay a point clear of second-placed Borussia Dortmund, who crushed VfB Stuttgart 6-1 on Friday.
(Reporting by Karolos Grohmann; editing by Toby Davis)
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