Miami appeared to have the victory within its grasp through the first eight innings, but the Braves made a late-game rally in the ninth.
With two runners on base and just one out, Dan Uggla brought in Jason Heyward. Then, Brian McCann doubled off the wall in left center field, which scored Uggla and Chipper Jones and tied the game at three runs apiece.
It took Reyes' hit to bring in Rob Brantly for the game-winning run.
Atlanta still holds a comfortable lead for the top National League Wild Card spot.
After a pair of scoreless innings, Gorkys Hernandez hit an RBI-double to score John Buck and put the Marlins up, 1-0. Three more scoreless innings went by before John Buck blasted a two-run shot for his 11th home run of the year.
Nathan Eovaldi pitched well through eight innings, allowing no runs and just four hits with five strikeouts. But relievers Steve Cishek and Mike Dunn had problems closing out the game.
Heath Bell was awarded the win for maintaining order in the 10th inning.
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