Tyson Foerster and Travis Konecny scored for Philadelphia (37-26-12). Samuel Ersson made 15 saves.
The Flyers are two points behind the Senators for the second wild card. They also are three points behind the New York Islanders for third place in the Metropolitan Division and play them at UBS Arena on Friday.
"I thought we played a pretty good game," Konecny said. "We'll just get on to the next game tomorrow. ... We've got a big game tomorrow, so just keep pushing."
DeBrincat put the Red Wings ahead 1-0 with a power-play goal at 18:49 of the first period. Kane found DeBrincat alone in the middle of the zone above the face-off circles, and he scored past Ersson's glove.
The Red Wings went 1-for-2 on the power play and 1-for-1 on the penalty kill. Special teams were a big emphasis for Detroit after Philadelphia went 2-for-2 on the power play and 2-for-2 on the penalty kill in the Flyers' 5-3 win at the Red Wings on March 28.
"You win the special teams battle on any given night, you have a much better chance," Red Wings coach Todd McLellan said. "We were down by two in Detroit last week and tonight to win it, maybe it was the difference."
Foerster, playing for the first time since Dec. 1, tied the game 1-1 at 31 seconds of the second period. A loose puck rolled to Foerster in the high slot, and he dragged it back and shot it past Gibson's blocker. The forward had missed the previous 49 games because of an arm injury.