The Brewers fell 4 ½ games behind Chicago in the NL Central, the farthest Milwaukee has been out of first place all season.
The Brewers took a 2-1 lead into the eighth, when Michael Taylor homered on the first pitch from rookie reliever Josh Hader (1-2). Difo bunted Hader's second pitch for a single and later scored on Turner's double.
Ryan Madson (5-4) pitched a scoreless seventh in his first outing since Aug. 12. He was reinstated from the disabled list on Friday.
Sean Doolittle pitched the ninth for his 14th save in 14 chances with the NL East-leading Nationals since being obtained from Oakland on July 16. With one runner on, Manny Pina hit a fly near the center field wall that Taylor caught to end the game.
It was just the second time that Hader had appeared in back-to-back games this season. He allowed two runs on three hits and only retired one batter.
Milwaukee rookie Brandon Woodruff, recalled from Triple-A Colorado Springs on Friday, allowed one run on two hits and a walk while striking out eight in seven innings. In four starts this year, he has a 1.52 ERA.
The teams traded runs in the fourth inning. Milwaukee took a 2-1 lead in the sixth when Stephen Vogt doubled and right fielder Jayson Werth bobbled the ball for an error, allowing Neil Walker to score from first.