Backstrom scored two second period goals and completed the natural hat trick early in the third.
Alex Ovechkin and Tom Wilson had two assists each for Washington, which has won four of its last five.
Tyler Johnson (with 4:24 remaining) and Matthew Carle (with 2:31 left) scored late in the third period to pull the Lightning within one, but Eric Fehr's empty netter sealed it.
It was Washington's second win this week against the Lightning, following Tuesday's 5-3 victory at Tampa Bay.
The Lightning, which was beginning a five-game road trip, has lost 3 of 4.
Ben Bishop made 24 saves for Tampa Bay, which had a first-period goal disallowed to due goaltender interference.
Holtby was stellar during the second period.
Mike Green (hooking) and Ovechkin (roughing) were both sent off at 6:49, giving the Lightning a 5-on-3 power play for two minutes.
Holtby made a series of saves, including one on a rebound try by Ryan Callahan when he reached back with his stick to pull the puck away just before it rolled across the goal line.
The Capitals took a 1-0 lead when Backstrom corralled a blocked shot in the slot and back-handed the puck in the net.
Backstrom has five points (three goals, two assists) in his last three games.
He made it 2-0 with 1:20 left in the period. Bishop made the initial save on Backstom, and Ovechkin's rebound try rolled free.
Backstrom pulled the puck back as it crossed the end line just left of the goal, and put it in.
He completed the natural hat trick at 2:17 of the third when he took a pass in the slot from Ovechkin and snapped the puck low past Bishop for his eighth of the season.
Backstrom's only previous three-goal effort came April 17, 2010 against the Canadiens in a first-round playoff game.
Tampa Bay was outshot 12-2 in the first period, but appeared to take a 1-0 lead at the 9:56 mark.
Meanwhile, Holtby and Callahan were in a pile to the right of the net with Washington's Brooks Orpik standing over them. After a review, Callahan was called for goaltender interference and the goal was disallowed.
NOTES: Washington won the season series 2-1. . Tampa Bay F Ondrej Palat (lower-body injury) missed his second straight game, and D Radko Gudas sat out due to a stomach virus. . Washington D Karl Alzner played his 323rd consecutive game, passing Kelly Miller for the third-longest streak in franchise history.