Backstrom scored with 42 seconds left in regulation to even the score and then banged home the game-winner two minutes into the extra session to keep Washington unbeaten at home.
The loss wasted a night of offensive fireworks by 41-year-old winger Teemu Selanne, who had two goals and two assists for the Ducks, who average just two goals a game.
"We played so well earlier in the game and then we started to self-destruct," Anaheim coach Randy Carlyle said.
"They're an offensive team that pounced on their chances, put pressure on us, and we didn't react very well," he said.
Alex Ovechkin, twice NHL Most Valuable Player, was on the Capital's bench during the frantic final minutes of regulation, a move that puzzled many of the 18,000 at sold-out Verizon Center.
"I thought other guys were better than him and there was just a chance other guys might score the goal," said Capitals coach Bruce Boudreau. "I've got to put out the guys that I think are going to score the goal.
"Ninety-nine per cent of the time Alex is the guy I think is going to score the goal. I just didn't think he was going to score the goal at that time tonight.
"You go with your gut feeling. I got lucky."
Ovechkin, who has five goals this season after a slow start, recorded an assist on Backstrom's game winner.
Washington, winners of six straight at home, fired 40 shots and Jonas Hiller, while his counterpart for the Capitals, Tomas Vokoun, saw only 15.
Anaheim raced to a 3-0 lead before goals by Joel Ward and Dennis Wideman trimmed the Capitals' deficit to 3-2 with a period to play.
The Ducks' Corey Perry stretched the lead to 4-2 on a power-play goal before Troy Brouwer countered for Washington with eight minutes left, setting the stage for Backstrom.
Washington, who snapped a two-game losing streak after opening the year with seven straight wins, have 16 points, three more than idle Florida in one of the NHL's tightest divisions.