Stutzle, who scored twice, tapped in a cross-ice pass from Drake Batherson on a rush for the winning goal. He has six goals in his past six games.
Michael Amadio extended his goal streak to three games, and Linus Ullmark made 20 saves for the Senators (7-5-3), who are 5-1-3 in their past nine games.
Matvei Michkov and Jamie Drysdale each scored, and Christian Dvorak had two assists for the Flyers (8-5-2). Samuel Ersson made 10 saves.
Stutzle gave the Senators a 1-0 lead at 5:14 of the first period when he one-timed a pass from David Perron over Ersson's right shoulder from just below the right face-off dot.
Amadio made it 2-0 at 6:19. He took a centering pass from Claude Giroux, skated in from the slot and beat Ersson glove side with a backhand shot.
Michkov cut it to 2-1 at 11:23 of the second period. He shook off a check from Jake Sanderson, drove to the slot and beat Ullmark over the glove with a snap shot.
Drysdale put a loose puck past a sprawling Ullmark at the edge of the crease to tie it 2-2 at 9:55 of the third period.
