His second RBI of the contest and second hit capped a two-run rally. He was credited with an RBI on a fielder's choice groundout in the sixth inning.
No other player had more than one hit for the Nationals, who swept a brief two-game series after getting just six hits.
The winner was reliever Saul Rivera (3-1).
Starter Shawn Hill yielded an earned run and just four hits in eight innings.
Kotsay's third hit gave Atlanta a 2-1 lead in the top of the 12th.
Both Escobar and Mark Teixeira had two hits for the Braves, who have lost five of eight against the Nationals this season.
The losing pitcher was Manny Acosta (0-1).