The Capitals held a 3-1 lead after one period only to see the Lightning rally in a bid to pick up what would have been their fourth win in five games.
Ondrej Palat tied the score with 5:30 to play and overtime was looming when Fehr deflected a shot from Mike Green to collect his second goal of the night and his eighth of the season.
Washington snapped a four-game losing streak
Fehr opened the scoring 7:04 into the game to create a 1-1 tie and Washington came up with additional first-period goals from Mikhail Grabovski and Marcus Johansson, both of which were produced on the power play.
Philipp Grubauer earned the win by making 33 saves. Anders Lindback gave up four goals on just 20 shots.