The Braves prevented Washington from what would have been its 90th win of the season and moved within 7 1/2 games of the Nationals in the NL East.
Atlanta held a 1-0 lead until Bryce Harper hit his 19th homer of the season in the sixth inning to tie it.
The Braves then scratched out the winning run against Sean Burnett (1-2).
Simmons reached on an infield single with one away in the ninth and moved to third on a base hit by Michael Bourn.
Desmond was charged with an error, making the run unearned.
Craig Kimbrel (2-1) picked up the victory after striking out the side in the ninth. Atlanta starter Kris Medlen struck out a career-best 13 batters in seven innings.