The Dodgers had struck out 20 times in the game before Puig came up with two outs in the 11th and blasted a pitch from Curtis Partch (0-1) deep into left-field seats for his 10th homer, boosting National League West-leading Los Angeles to its 26th win in 32 games.
Brandon League (6-3) earned the decision with two innings of scoreless relief after starter Chris Capuano and three relievers had combined to blank the Reds over nine innings.
With the victory and Arizona's loss to San Diego earlier Sunday, the Dodgers moved 2 1/2 games in front of the Diamondbacks atop the division standings.
Reds starter Tony Cingrani held the Dodgers to one hit over seven shutout innings, striking out 11.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys