Gradkowski was 14-for-21 for 178 yards and one interception.
Cadillac Williams ran for 122 yards on 27 carries. Joey Galloway scored a touchdown on four receptions for 64 yards, and Matt Bryant kicked two field goals for the Buccaneers, who broke a three-game losing streak.
In his first NFL start, Jason Campbell completed 19 of 34 passes for 196 yards and two touchdowns for Washington. Lydell Betts, playing for the injured Clinton Portis, made five catches for 48 yards, while Chris Cooley and Todd Yoder caught touchdown passes for the Redskins, who have dropped five of their last six games.
Washington, which relies on its ground game, gained only 64 net rushing yards.
Portis is out for the season with a broken right hand.