Ankiel singled off Burnett with one out in the ninth and Clippard came on to walk David Ross. Omar Infante then struck out, giving Clippard a chance to get out of the inning, but Heyward ripped a single up the middle on a 3-0 pitch.
The loss went to Burnett (0-7), who has appeared in relief 55 times this season without a victory.
Atlanta has won 21 times in its final at bat. The Braves have captured four in a row and remained 2 1/2 games in front of Philadelphia atop the National League East.
Closer Billy Wagner came on in the top of the ninth and struck out the side to get the victory. He improved to 7-2 along with his 29 saves.
Atlanta starter Tim Hudson ran his scoreless innings streak to 19 before giving up a run in the third.