Newly acquired second baseman Anderson Hernandez started the decisive rally by drawing a leadoff walk off Vladimir Nunez (0-2) and went to second on a passed ball by catcher Brian McCann.
Cristian Guzman lined an infield single off the glove of first baseman Martin Prado, sending Hernandez to third. Ryan Zimmerman was then intentionally walked to load the bases.
Lastings Milledge struck out and Ronnie Belliard lined out before Dukes took a low and outside 3-2 pitch to force in the winning run.
Steven Shell (1-1) pitched an inning for the win.
Hernandez and Jesus Flores each drove in three runs for the Nationals, whose winning streak is their longest since a six-game run from July 31-Aug. 5, 2007.
Atlanta's Chipper Jones, who entered the game as the majors' leading hitter, homered off Charlie Manning leading off the seventh, his 20th of the season. He joins Hall of Famer Eddie Mathews as the only players in baseball history to start their careers with 14 seasons of at least 20 home runs.
Belliard added two RBI and Zimmerman three hits for Washington, which has won nine of 13 games against Atlanta this season.
McCann and Prado also homered for the Braves, who have lost 14 of 19. Jones, McCann and Prado each drove in two runs.
Atlanta fell to 0-21 in one-run road games this season and is 0-28 dating to last year.
The Braves used four doubles and an RBI single by Prado to go up 4-0 in the third. After Josh Anderson doubled for the Braves' first hit against Nationals starter Jason Bergmann, Jones, McCann and Omar Infante hit RBI doubles. Prado's run-scoring single to the alley in right-center capped the inning.
Hernandez's two-run single made it 4-2 in the fourth.
McCann lined Bergmann's 3-2 pitch over the right-field wall leading off the fifth. Two batters later, Prado's second home run put Atlanta up 6-2 and chased Bergmann.
Bergmann went 4q innings and gave up six runs, eight hits and three walks.
Braves starter Jo-Jo Reyes appeared poised to get his first victory in 10 starts since June 13, but couldn't get through the fifth inning, when the Nationals scored five times to take the lead. Reyes was lifted after yielding singles to Zimmerman and Milledge.
Buddy Carlyle relieved and gave up an RBI double to Belliard before walking Dukes to load the bases. Flores cleared them with a game-tying, three-run double to the wall in center.
Hernandez then put Washington ahead 7-6 with a two-out, RBI single to center. Belliard added a run-scoring single in the sixth for a two-run lead.
Reyes worked four-plus innings, giving up four runs and eight hits.
Atlanta tied the game in the eighth. Pinch hitter Greg Norton and Anderson singled before Gregor Blanco's RBI double to right. Anderson was thrown out at home trying to score.