Houston's Fernando Rodriguez (1-8) walked the bases loaded with none out in the 10th, prompting the Astros to play the infield in.
Hart then stroked a sharp grounder that got past third baseman Matt Dominguez. Shortstop Jed Lowrie fielded it, but couldn't make a throw as Nyjer Morgan crossed the plate with the go-ahead run.
Rickie Weeks (3-for-5, two RBI) added a run-scoring single to cap the inning and send the Brewers to a victory in the rubber match of their three-game series.
The Brewers forced extra innings when Ryan Braun singled home the tying run in the eighth.
Jose Altuve, J.D. Martinez and Chris Snyder knocked in runs for Houston.