Matthew Tkachuk scored in the four-round shootout, while Sean Monahan, Johnny Gaudreau and Andrew Mangiapane were unable to find twine.
Drake Batherson had the 1-on-1 winner for Ottawa, beating Jacob Markstrom.
Mark Giordano, Gaudreau and Noah Hanifin scored for the Flames in regulation, while Markstrom made 19 saves.
Matt Murray got the start again for the Sens and had 31 stops.
Ryan Huska served as the bench Sunday for the second straight game, with new head coach Darryl Sutter finishing up COVID protocols and slated to arrive at the 'Dome Monday.
Prior to game - on a night the Flames paid tribute to International Women's Day (more on this below) - there was also a moment of silence to honour the great Walter Gretzky, who passed away Thursday night.
The Flames got an early powerplay 1:06 in when Derek Ryan - who usually plays centre but started the tilt on the wing with Mohanan and Gaudreau - was tripped while trying to tip a puck out of the Calgary zone up by the blueline for a potential breakway attempt.
Calgary wasn't able to convert but got another crack on the man-up just under four minutes later, again stymied.
Ottawa opened the scoring on a weird one from Connor Brown, who snapped a wrister from up near the blueline boards that looked to deflect off Chris Tanev and beat Markstrom five-hole at 13:48.
The Sens got a powerplay not long after which was cut short when Mangiapane was tripped on a 2-on-1 shorthanded rush.
On the ensuing four-on-four, Ryan Dzingel tapped home a feed from Chris Tierney on an Ottawa 2-on-1 to double the lead at 15:56.
Brady Tkachuk streaked in on a breakaway just 1:08 into the second period but Markstrom came up big to get his shoulder on the shot and deflect it high.
Zac Rinaldo and Austin Watson dropped the mitts just over a minute later for a quick fight right beside the penalty box, both then sitting down for five minutes.
Giordano scored his third goal of the year - and 496th career point - at 3:21, taking a pass from Tkachuk and firing his own wrister through traffic that beat Murray five-hole. Nikita Nesterov got the other helper.