Rollins had two RBI singles and a walk for the Phillies, who began the day two games behind the first-place New York Mets in NL East.
J.C. Romero, Chad Durbin and Brad Lidge finished the six-hitter. The Nationals have been shut out 20 times, seven more than any other major league team.
Washington's winning streak was its longest since a 10-game run in 2005.
John Lannan (8-13) took the loss, giving up two runs and five hits in six innings. He walked two and struck out four.
Nationals catcher Jesus Flores was carted off the field in the third inning after Philadelphia's Chase Utley barreled into him on a play at the plate. X-rays showed no obvious signs of a fracture and an MRI was pending, the team said.
The collision came as Utley was attempting to steal home with two outs. With runners on first and third, Lannan made a pickoff throw to first and Utley broke for home.
Utley arrived at the plate just after the throw from first baseman Ronnie Belliard and crashed hard into Flores. After he tagged out Utley, Flores stayed on the ground near home plate for several minutes as his left leg was attended to by trainers and medical personnel.
Eventually, Flores was placed on a stretcher and carted off the field. He was replaced by Wil Nieves.
The collision was the second play at the plate for Flores on Tuesday.
Earlier in the third inning, Rollins singled home Carlos Ruiz from second. The throw from left fielder Willie Harris was in time but slightly to the right of Flores, who missed the ball as he tried for a sweep tag on Ruiz. Rollins, who went to second on the throw, scored on Utley's single.
The Phillies tacked on two more runs in the seventh, the first on another RBI single by Rollins. Ruiz then scored as Rollins was caught in a rundown between first and second during an inning-ending double play.