Braun, leading off the 11th, blasted a long drive to center field off Colorado reliever Matt Lindstrom for his 28th home run of the season to end the game, but the performance of the Brewers pitching staff was key.
"All the credit tonight goes to our bullpen, to (Zack) Greinke for battling through," Braun told reporters. "I don't think we really deserved to win but good teams find a way to win games sometimes that they shouldn't.
"We were fortunate to win."
Milwaukee starter Greinke struck out nine batters and allowed one run over five innings before leaving with his team trailing 1-0.
Kameron Loe got the win by working the top of the 11th, where he got the final out despite two Colorado runners in scoring position.
The Brewers used five relievers in the final six innings to keep the Rockies (69-78) off the scoreboard.
Colorado had chances to score more runs early, but Mark Ellis was thrown out at home plate in the first inning.
He was also tagged out at home in the third after trying to score on a pitch that got away from catcher Jonathan Lucroy, only to watch Greinke make a stunning grab and flip of the ball to home plate for the out.
(Editing by John O'Brien)