Kelly snagged a toss by Jason Campbell, who was intercepted twice in the fourth quarter and lost a fumble earlier in a game, in which the Redskins had six turnovers.
Tampa Bay (7-4) played most of the game without quarterback Jeff Garcia, who left in the first quarter with a back injury and returned briefly in the fourth.
Bruce Gradkowski connected on 19 passes for 106 yards in Garcia's absence.
Campbell threw for 301 yards and a touchdown.
Shaun Suisham kicked two field goals for Washington (5-6).
Notable deaths of 2014 [PHOTOS]