Philadelphia trails the Boston Bruins by six points for fourth place in the eight-team East. The top four teams in the division will qualify for the Stanley Cup Playoffs.
Sheary gave the Capitals a 1-0 lead when he skated below the left face-off circle and scored on a wrist shot from a sharp angle at 7:07 of the first period.
Couturier tied it 1-1 at 8:45 when he deflected James van Riemsdyk's shot.
Wilson put Washington ahead 2-1 when he scored a power-play goal from the bottom of the right circle on the rush at 11:32.
Backstrom took a cross-ice pass from Ovechkin during a power play and scored on wrist shot from below the right circle at 16:25 to make it 3-1.
The Capitals were 3-for-3 on the power play after going 3-for-6 their previous game, an 8-1 win at the Bruins on Sunday.
"Guys are moving the puck pretty quick. I think all options are in play," Washington coach Peter Laviolette said. "You see that man in the middle open up quite a bit. The shots from the point. I think once you start, if the far side gets taken away, you have to find opportunities on the interior, and I think that the guys have done a really good job of that."
Carl Hagelin increased the lead to 4-1 with a wrist shot from the right hash marks at 18:21.
"Obviously, I didn't like at all the way we came out," Flyers coach Alain Vigneault said. "Not the way you give yourself a chance against such a strong opponent. It wasn't good enough. We put ourselves in a hole."
The Flyers were 10-for-10 on the penalty kill their previous three games.
"Tonight, I think the PK didn't get the job done, and it's tough for the momentum of a game when they're 2-for-2 (on the power play) in the first," Couturier said. "We've got to be better than that in that area of the game. But all around, too, we need to be much better, solid. It's tough. It's not fun this year, I guess."
NOTES: John Carlson had two assists to reach 400 in the NHL in his 800th game. He is the second defenseman in Capitals history to play at least that many games (Calle Johansson, 983). … Backstrom passed Daniel Alfredsson for fourth in assists by a Sweden-born player in NHL history with 714, behind Mats Sundin (785), Henrik Sedin (830) and Nicklas Lidstrom (878). ... Washington forward Michael Raffl, acquired from Philadelphia in a trade Monday, did not play.