Campbell connected on 20-of-31 tosses for 231 yards and Clinton Portis ran for 121 yards for the Redskins (3-1).
Suisham kicked a 29-yarder with 3:22 to play that sent the Redskins ahead by nine points. Quarterback Tony Romo then took Dallas on a drive that led to a touchdown at 1:42 but Washington got the ball on an onside kick and ran out the clock.
Santana Moss was Washington's leading receiver, catching eight passes for 145 yards.
Romo was 28-of-47 for 300 yards, three touchdowns and an interception.
Dallas running back Marion Barber was held to 26 yards on eight carries.