Espinoza reached on a one-out fielder's choice against Cristhian Martinez (5-3), and then advanced to third when Kurt Suzuki's chopper over the mound evaded three Atlanta infielders for an infield hit. He took off for the plate when pinch-hitter Chad Tracy sent a grounder to Uggla.
The Braves second baseman fielded it but dropped the ball before he could throw home and Espinoza crossed the plate with the game winner.
The late run made a winner of Craig Stammen (6-1) -- the Nationals' eighth pitcher -- who tossed two scoreless frames for the National League East leaders.
The victory gave Washington a six-game lead over second-place Atlanta and put the Nationals 30 games over .500 (76-46).
Jason Heyward socked a two-run homer and added a run-scoring single for the NL wild-card-leading Braves.