Strasburg (6-9) dominated the reeling Phillies for his first win since July 7 against San Diego. The Nationals had lost five straight games started by the right-hander.
The Phillies never had a runner in scoring position during their 11th straight road loss. Philadelphia has lost nine of 11 overall.
Washington scored two runs in the fourth and three in the fifth against Kyle Kendrick (9-9), who surrendered 11 hits over 5 1-3 innings.
Werth, who had a go-ahead, two-run homer among his three hits Saturday, delivered an RBI single in the first inning to give Washington a 1-0 lead.
Steve Lombardozzi also had three hits, including an RBI single.
The sweep denied Phillies manager Charlie Manuel the 1,000th win of his career.
Philadelphia had not been swept by the Nationals-Expos franchise since losing a four-game series from August 25-28, 2003, while in Montreal.
In between wins, Strasburg went 0-3 with 4.93 ERA in five starts. Two of Strasburg's six victories on the season have come against the Phillies.
Strasburg's biggest stress came starting the second inning. He winced after throwing a second straight ball to Domonic Brown, leading to a visit from the Nationals' training staff.
After walking Brown on four pitches - the only walk he issued - Strasburg struck out Darin Ruf and Cody Asche on his way to retiring 14 of the next 15 batters. The right-hander has allowed two or fewer runs in 11 of 13 home starts.
Making his fourth start against the Nationals this season, Kendrick allowed six runs, four earned. He has allowed 14 earned runs over his last 13 1-3 innings.
Wilson Ramos and Denard Span each had RBI singles in the fourth inning.
Chase Utley's throwing error in the fifth allowed two runners to score. Lombardozzi tacked on an RBI single for the Nationals' third run of the inning.
Kevin Frandsen had two of the Phillies' four hits.