Stutzle won it when he took a pass from Brady Tkachuk, skated past Flyers defenseman Travis Sanheim in the slot and scored with a backhand around the left skate of Dan Vladar.
Nick Cousins also scored for Ottawa (28-22-7), which won for the fifth time in six games. James Reimer made 15 saves.
Reimer started the game because Linus Ullmark was unavailable because of illness after taking part in the morning skate. Hunter Shepard was recalled from Belleville of the American Hockey League and arrived during the first period to serve as the backup.
Jamie Drysdale scored for Philadelphia (25-20-11), which lost for the 12th time in 15 games (3-8-4). Vladar made 25 saves.
It was the final game for both teams before the break for the Olympic Winter Games Milano Cortina 2026.
The Flyers tied it 1-1 when Drysdale scored at 18:46 of the third period. With Vladar pulled for an extra attacker, Drysdale took a cross-ice pass from Bobby Brink and scored from the top of the left face-off circle.
Cousins gave the Senators a 1-0 lead at 9:55 of the second period. Shane Pinto brought the puck out from behind the net and had his shot from low in the right circle stopped by Vladar with the right pad before Cousins skated into the crease to bang in the rebound.