Ryan Zimmerman, in a 2-for-14 slide coming in, had three hits for Washington and Adam LaRoche added a pair.
Bryce Harper hit an RBI groundout in the first following Denard Span's leadoff double, and Harper came home from second when LaRoche singled and centerfielder Austin Jackson let the ball get past him for an error. Ian Desmond followed with a run-scoring single for a 3-1 Washington lead.
Fister (4-1) had an RBI single in the second off Haren (4-3), but the Nationals made it 5-1 in the bottom half on consecutive run-scoring singles by Zimmerman and LaRoche.
Haren (3-3) allowed four runs and nine hits in six innings.
After falling behind 5-1 after two innings, the Tigers pulled to within 5-4 on Matt Tuiasosopo's pinch-hit three-run homer in the sixth. Leyland sent Tuiasosopo, a right-hander, up to pinch-hit against the right-handed Haren.
"Right-handers have a little have better numbers off Haren over his career and we were looking for a quick strike and we got it," Leyland said.
Washington's three relievers had identical lines, each giving up a hit and striking out a batter in an inning. Soriano converted his 12th save in 13 chances.
NOTES: Tuiasosopo's home run was the first by a Tiger pinch hitter since Ryan Rayburn at the Chicago White Sox on Sept. 14, 2011. . Fister's RBI was the first by a Tigers pitcher since Rick Porcello had two on June 12, 2009.The game was a makeup of Tuesday night's rainout. Rain fell during the second and third innings Thursday but play was not halted. ... Leyland has managed at every current big league ballpark other than Marlins Park, which opened in 2012. ... Ross Detwiler (1-3, 2.50) will face Cubs RHP Jeff Samardzija (1-4, 3.09) on Friday night in Washington. ... Max Scherzer (4-0, 3.43) starts at home Friday, opposed by RHP Cleveland's Corey Kluber (2-1, 3.06).