Felix Sandstrom made 43 saves in his NHL debut for the Flyers (13-12-6), who extended their point streak to seven games (5-0-2).
"Honestly, we should probably spend 30 seconds to talk about their goalie," Reimer said. "Hats off to him."
Morgan Frost gave the Flyers a 1-0 lead at 15:23 of the first period with a shot in close.
Couture tied it 1-1 with a shorthanded goal at 5:20 of the second period when he stole the puck from Flyers defenseman Keith Yandle and scored on a breakaway.
Burns gave the Sharks a 2-1 lead at 7:20 with a slap shot from between the circles.
"We dug in and came from behind and did a lot of good things," Sharks coach Bob Boughner said. "It could've been nice to find some puck luck earlier. But we found a way to come back."
The Sharks killed a 5-on-3 Flyers power play for 1:56 beginning at 7:33 of the third period.
"Tonight I'm going to focus on the positives," Flyers coach Mike Yeo said. "I definitely saw some tired mistakes out there. When you're tired sometimes the execution isn't quite there, sometimes your pace isn't quite there. I thought the will and the compete was there all night.
"Obviously I think we have to continue to focus on trying to build our game. That's not the game we want to be playing where we're spending that much time in our own zone. We don't want to be a team that's getting outshot, outchanced."