Late in overtime, Flyers goalie Steve Mason stopped Milan Michalek as he came in on a two-on-one, keeping the game alive.
Like Dallas in its 5-1 win over the Flyers on Saturday, the Senators had numerous odd-man rushes and quality scoring chances.
"Every time they had the puck, they were jumping, and [defenseman Erik] Karlsson was on the rush the whole time," said Claude Giroux, who couldn't convert a great scoring chance with 48 seconds left in regulation and shot wide in the shootout. "We have to do a better job of reloading and playing good defense."
Kimmo Timonen's first goal in the last 10 games, a shot from near the left boards, tied the score at 4-4 with 9 minutes, 2 seconds remaining in the third period.
With 5:38 to go, Ottawa goalie Craig Anderson made a sensational glove save on Jake Voracek's backhander as he came in alone, keeping the score tied.
Kyle Turris, who lost a fight to Couturier (the first bout of his career) earlier in the game, scored from above the right circle with 11:39 left to give the Senators a 4-3 lead. Mason appeared to be screened by Ottawa's Jared Cowen.
Mason had not allowed more than three goals in any of his 27 games with the Flyers before surrendering four in the loss in Dallas.
The third-period breakdown ruined a breakout night by three Flyers who ended long goal-scoring droughts: Michael Raffl, Voracek and Schenn.
Raffl's goal was the first of his NHL career and ended a 17-game drought.
Voracek hadn't scored in his last 12 games, and Schenn ended a 23-game drought for the Flyers (13-14-3), who are 2-2-1 on the six-game road trip that ends Wednesday in Chicago.
Last month, the Flyers posted 5-0 and 5-2 wins over the Senators, who had been 2-12-2 when trailing heading into the third period.
In the shootout, Spezza, moving deliberately, beat Mason after a series of dekes.
"It was something we saw in the pre-scout," Mason said of Spezza almost coming to a stop before making a move. "I held my ground there, but great hands by him and a good move."
The Senators, now 1-4 in shootouts this season, had tied it at 3-3 on Clarke MacArthur's second goal of the night with 17:39 left in the third period.
Raffl had come close to scoring in recent games, and he finally cashed in after some terrific work by Simmonds, who went wide before firing a shot. Raffl scored on a follow-up of a Simmonds shot, giving the Flyers a 1-0 lead with 11:42 left in the opening period.
"It is such a relief," said Raffl, an Austrian who is making the jump from a lower-level Sweden league, after his first goal in 18 games. "I've had so many scoring chances and it finally goes in."
Raffl finished with a career-best two points.
"The guys battled hard the whole game," coach Craig Berube said. "I don't think we generated enough early in the game, but we picked it up later on. We were too much in our own end in the first half" of the game.