Brady Tkachuk dug the puck out at the left boards and sent a pass to Cozens in the slot, where he buried a wrist shot past Devin Cooley's blocker to put Ottawa up 2-1 with his fourth goal in three games.
Lars Eller scored for the Senators (30-22-9), who are 3-0-2 during the streak and 7-1-2 in their past 10 games. Linus Ullmark, who is 8-0-3 in his past 11 starts, made 19 saves.
Martin Pospisil scored the lone goal for the Flames (24-30-7), who have lost four straight (0-3-1). Cooley made 33 saves.
Pospisil gave Calgary a 1-0 lead at 17:04 of the first period, collecting a pass from Matvei Gridin and sending a wrist shot on net that deflected in off Ullmark's right arm. It was Pospisil's first goal of the season in his 10th game, and it came in his first game back after being a healthy scratch for the past two.
Eller tied it 1-1 at 9:35 of the second period when he one-timed a Ridly Greig cross-ice feed to the glove side from the top of the right face-off circle.
Tim Stutzle scored a short-handed, empty-net goal at 17:59 of the third period to make it 3-1, and Shane Pinto also scored into an empty net at 19:46 for the 4-1 final.
