Brady Tkachuk, Drake Batherson and Alex DeBrincat also scored for Ottawa and Stutlze had assists on all three goals. Mads Sogaard, appearing in his fourth NHL game, made 34 saves.
''A lot of pucks bobbling in the slot and on the power play, too,'' Stutzle said. ''We had some good looks and the puck bobbled every time we wanted to shoot. . I got really mad there a couple of times, but in the end we figured it out. We did a great job staying with it.
The 22-year-old Sogaard got the start with Cam Talbot and Anton Forsberg sidelined. He had 10 saves in the first period, 15 in the second, eight in the third and one in overtime.
''It's awesome to be here and awesome to get a win in front of our fans and you know it's my first win here,'' Sogaard said. ''I lost the last time I played against Winnipeg so it was a very special moment and something I won't forget.''
Senators coach D.J. Smith gave Sogaard plenty of credit.
''He looked really calm,'' Smith said. ''I mean he's big, obviously, pucks hit him and he made the saves he had to make. They had plenty of chances and he looked good.''
Dillon Dube had two goals and Tyler Toffoli had a goal and an assist for the Flames. Jacob Markstrom finished with 21 saves.
''We had chances throughout the entire game to essentially put them away, and we didn't,'' Toffoli said. ''We let them hang around. They're a team with a lot of skill and can do what they did.''
The Senators trailed 3-1 late in the third period before Batherson scored on the rebound of a shot by Stutzle with 2:14 left and DeBrincat tied it 46 seconds later. Both goals came with Sogaard pulled for an extra skater.
''There's four, five minutes left and you see a few people leaving and before you know it, it's tied up,'' Batherson said. ''It's awesome. We haven't scored many 6-on-5 goals since I've been here and the place went nuts so it was great to get the win.''
The Senators got off to a bit of a slow start, but were first on the scoreboard when Stutzle fed Tkachuk a great pass to spring him loose for a breakaway goal 6:10 into the first period. It was his 21st goal of the season.
Dube tied it on a power play when he deflected Noah Hanifin's shot past Sogaard with 5:13 left in the opening period.
Toffoli put Calgary ahead at 5:55 of the second - just seconds after a power play expired - when he beat Sogaard on the glove side with his 22nd.
Dube put the Flames up 3-1 with his second of the night and 15th of the season by tipping Toffoli's shot past Sogaard.
''I thought we dominated the game,'' Flames coach Darryl Sutter said. ''I thought we totally dominated puck possession. Whether it's the next goal or the next save, I thought we were in total control of the game.''
Ottawa LW Austin Watson returned to the lineup after being a healthy scratch the last three games. ... Talbot is expected to return in seven to 10 days, while Forsberg, who injured his right and left MCL, will be out two to three months.
Flames: Host Detroit on Thursday night.
Senators: At the New York Islanders on Tuesday night.
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