Andrelton Simmons opened the third inning of a scoreless game with a double, but was erased at third on a fielder's choice grounder off the bat of pitcher Kris Medlen.
Medlen (11-12) gave up six hits and struck out six over seven scoreless innings to get the win. Craig Kimbrel retired the Indians in order in the ninth for his 43rd save.
The Cleveland run came on an eighth-inning homer by Lonnie Chisenhall.
Atlanta moved closer to a division title by winning its fourth straight. The Braves are 13 games in front of second-place Washington in the National League East.