The contest remained scoreless over eight innings until Hechavarria smacked a leadoff triple against Jordan Walden (4-2), then came home when Walden uncorked a wild pitch with Ed Lucas at the plate.
Mike Dunn (3-3) recorded the final two outs in the eighth to gain the victory and Steve Cishek notched his 24th save for the Marlins, who broke a six-game skid and a 22-inning scoring drought.
Miami starter Nelson Eovaldi yielded just one hit over seven scoreless innings, walking three and striking out eight.
Alex Wood matched him for six innings, allowing only two hits while walking one and striking out seven for National League East-leading Braves, who fell for the first time since July 25.
Celebrity Couples of 2014 [PHOTOS]