Mike Webber fired a slapper that knocked off Michael Leighton's pad. Ennis swooped in from the right side and scored his second goal of the game.
Game 6 is Sunday in Buffalo.
The Sabres survived after blowing a 3-0 and chasing Flyers starting goalie Brian Boucher.
Leighton was flawless until overtime.
Thomas Vanek and Marc-Andre Gragnani also scored in the first period for the Sabres.
James van Riemsdyk, Andrej Meszaros, Danny Briere scored for the Flyers.
The Flyers, the No. 2 seed in the East, have no Jeff Carter, no Chris Pronger and are on their third goalie of the series. Their biggest problem could come Sunday where the Sabres are poised to send the defending Eastern Conference champions home for the offseason.
Ryan Miller made 36 saves for Buffalo.
Staked to the early lead, it seemed over for Flyers. Miller has 1-0 victories in the series and appeared to again stump the Flyers. Not for long.
After van Riemsdyk and Andrej Meszaros scored in the second, Briere stuck it to his former team when his backhander off a behind-the-net feed from Mike Richards tied it at 3.
The rally ended there.
Ennis pounded the winner past Leighton, who made only his second appearance for the Flyers this season.
Stuck in a 3-0 hole, the Flyers felt at home.
After all, last season they trailed Boston 3-0 in the Eastern Conference semifinals and won the series. The Flyers won Game 7 after trailing 3-0.
Like last year's Game 7 in Boston, van Riemsdyk who struck first.
With Miller a few feet in front of the crease, JVR punched the puck in the empty net to end Miller's shutout streak.
Meszaros then scorched a liner from the point to make it 3-2.
Flyers fans busted out the derisive "Miller! Miller!" chants. The Flyers outshot the Sabres 15-5 in the second and appeared in control.
Hard to believe considering the damage Buffalo inflicted in the first.
For the second time in the series, the Sabres chased a Flyers goalie after taking a 3-0 lead. In Game 2, it was rookie Sergei Bobrovsky. On Friday, Boucher was yanked after allowing three goals on 11 shots.
Leighton got the call, capping one of the improbable rides of any recent Flyer. He started for the Flyers in the Stanley Cup finals, signed a two-year deal in the summer, then hurt his back and spent all but one game in the minors.
Leighton played his first game in Philadelphia since a Jan. 21 start for the Adirondack Phantoms, the Flyers' AHL affiliate.
Improbably, he'll likely start Game 6.
Boucher allowed two of the softest goals of the season only 1:27 apart early in the first. Ennis and Vanek scored after Boucher let the pucks knock off his leg and into the net.
Gragnani banked the puck off the post past a surprised Boucher on a power play for a 3-0 lead late in the first and the Sabres were clearly in control.
That "Booooosh!" cheer that sounded so sweet in Game 3 certainly had a different spin with the "sh" knocked off.
That was all for Boucher. Boucher knew he blew his shot to stick as the No. 1 goalie. He whacked a row of sticks with his goalie stick, kicked the sticks and tossed his helmet inside the tunnel leading to the locker room.
Leighton stopped the final 18 shots in regulation and he might just yet bail out the Flyers again in the playoffs.
The Flyers played without winger Carter and defensman Pronger. Carter was knocked out of Game 4 after he hurt his right knee in a collision, and Pronger, who appeared close to a return, is still out with a broken right hand.
The Sabres lost forward Jason Pominville and defenseman Jordan Leopold in the first period.
Pominville, who scored the only goal in Game 4, slammed down his stick and helmet and hopped on his right leg off the ice. He appeared to get sliced in the back of his leg after a collision in the corner of the Sabres' zone.
Leopold, who did return, appeared to get cut across his chin and was splitting blood on the bench.
NOTES: Leighton won his only start for the Flyers this season. ... A puck shattered a pane of glass during warmups.
