Alex Ovechkin and Nicklas Backstrom each had a goal for the Capitals (21-11-8), and Mike Green had two assists. Goaltender Braden Holtby, playing in his 23rd straight game and making his 17th straight start, made 30 saves.
The Flyers played the final two periods without defenseman Nicklas Grossmann because of an upper-body injury sustained with 1:41 left in the first period when he was hit into the boards by the Capitals' Tom Wilson. Flyers general manager Ron Hextall said he expected to have an update on Grossmann's condition Friday.
The Flyers capitalized after Matt Niskanen was sent to the penalty box for hooking at 1:11 of overtime. Claude Giroux sent a pass from low in the Washington zone to Voracek above the circles, and Voracek's shot deflected off Backstrom's stick and past Holtby.
"I knew if I was going to keep shooting, it had to go in eventually," said Voracek, who has 16 goals. "Got a lucky bounce, but it was a big goal."
It was the second straight game the Flyers were able to grind out a victory; they beat the Ottawa Senators 2-1 in a shootout Tuesday. Philadelphia was 2-17-6 entering the week when scoring three goals or fewer.
"I think it's important. Obviously we need to find a way to play in these low-scoring games," Schultz said. "I think we have showed that and stuck with it these last two and have found ways to win games. We have to build on that as we move into the second half."
Each team had a chance to take the lead on a power play late in regulation. The Capitals got a man advantage at 13:21 when the Flyers were called for too many men on the ice, and Philadelphia got a chance with 2:54 remaining when Washington was called for too many men.
The Capitals didn't get a shot on goal on their power play, but the Flyers got six, including a four-shot flurry when the puck found its way into the crease early in their man advantage.
It was the overtime penalty that had Capitals coach Barry Trotz upset. Of the Capitals' five overtime losses, four have come on power-play goals.
"You think about the last six minutes of the game, we had a chance on the power play and we didn't execute," he said. "We took a penalty at the end and we killed that off, then we took another penalty. That's got to be about the 10th time it seems that we've taken penalties in overtime. Just has to stop. ... It's self-discipline. Every time it's happened, sometimes there's an accidental stick or whatever, but most times it's a reaction or losing position. It's got to stop. They have to stop it. Once the guy is on the ice he has to do it. You're just hurting your teammates."
The Capitals also have noticed the disturbing trend.
"Things happen once, twice ... when we're making a habit of it, a trend of it, it's frustrating," Holtby said. "The way the NHL is now, overtime and shootout wins are necessary. We've got to give ourselves a better chance."
Despite picking up three of four points in their back-to-back -- the Capitals defeated the Toronto Maple Leafs 5-2 on Wednesday -- Trotz said he's unhappy with the way the games went on the short road trip.
"There's games we win and I'm not happy; there's games we win and I'm happy," Trotz said. "... I don't like the way we've played the last two nights. A lot of inconsistencies in our game we didn't like.
"Too many passengers on back-to-back nights here. Can't win against a team (the Flyers) that's trying to get back in it. They got a lot of urgency in their game. I thought they were sharper."
The Flyers felt sharper throughout the game, coming back twice after Capitals goals.
Backstrom tipped Alzner's point shot at 3:08 of the third period to give the Capitals a 2-1 lead; the goal was upheld as video review was inconclusive in determining whether Backstrom's stick was at or below the level of the crossbar when he deflected the puck into the net. But Umberger tied the game at 4:22 when Schultz's point shot took a sharp bounce off the end boards past Holtby and went right to Umberger, who scored into a nearly empty net.
The Flyers also got a lucky bounce on their first goal. After Ovechkin's shot from the left faceoff circle at 3:08 opened the scoring, Schultz's shot from the point bounced off Read and Green battling in the slot and right to an unmarked Couturier, who scored his 10th goal.
"Sometimes you're going to have a game where things aren't going to go your way all the time," Voracek said. "We didn't score many goals, but that was a huge one today."
Ovechkin's goal, his 21st, extended his goal streak to five games. It's his longest since he scored in five straight games from March 17-24, 2013.
"I think we started where we wanted," Ovechkin said. "We moved well. We scored on power play. Then we stopped playing."
The Flyers know this is how they have to keep playing if they have any hope of climbing back into the Stanley Cup Playoff race.
"I like the mental state of the team right now," Flyers coach Craig Berube said. "They're positive, and it's important to be positive. It's important to stick with it. This time of the season right now you have to win hockey games. It's a grind ... these teams are fighting for the playoff spot. It's going to be tight hockey, and I think right now we're sticking with it, doing a good job sticking with it, staying positive, and that's what you have to do."
Follow Adam Kimelman on Twitter: @NHLAdamK