The Nationals trailed 3-1 heading into the ninth but Alex Cora kept Washington's chances alive with a two-run single off Craig Kimbrel. Washington put a runner on second with one out in the 10th without scoring, but made the most of its chances one inning later against Scott Linebrink (0-1).
Ivan Rodriguez was hit by a pitch with one out and Cora singled Rodriguez to third. Desmond followed by a double to right that scored both runners and gave Washington a 5-3 lead.
That brought up Werth, who hit his sixth homer of the year.
Atlanta did not put a runner on base in either the ninth or 10th innings and then managed only a leadoff double in the 11th while Tyler Clippard was working in a non-save situation.
Drew Storen (3-1) worked the 10th as the fifth of six Washington pitchers to get the win.
The Braves used seven pitchers, who combined to give up 13 hits and walk five batters.