The Nationals seemed on their way to a 7-5 victory in the ninth but Jason Heyward extended the contest with a two-out, two-run homer off Washington closer Rafael Soriano -- his second home run of the game.
The score remained knotted until the 15th, when LaRoche led off the inning against Kris Medlen (10-11) with a long blast high into the right field stands for the decisive blow.
Ryan Zimmerman also homered while Wilson Ramos and Jayson Werth each drove in two runs for Washington, which lost starting pitcher Stephen Strasberg to an ejection for throwing at Braves batters in the second inning.
Chris Stammen (6-5) notched the victory with three innings of shutout relief and Dan Haren, the last of nine Washington pitchers, earned his first career save as the Nats evened their three-game series.
Celebrity Families of 2014 [PHOTOS]