Drake Batherson scored the deciding goal in the fourth round of the shootout. Tim Stutzle scored in the first round for Ottawa, and Matthew Tkachuk scored in the second for Calgary.
Mark Giordano, Johnny Gaudreau and Noah Hanifin scored, and Jacob Markstrom made 19 saves for the Flames (11-12-3), who came back from down 3-1 in the third period but fell to 3-6-2 in the past 11 games.
"It stings right now," Giordano said. "We know how important these points are. In the third, we had a great push, but we've got to find a way to get two points in games. That's the bottom line now."
Calgary fired Ward on Thursday after a 7-3 win against Ottawa, replacing him with Darryl Sutter. Assistant Ryan Huska coached the Flames for the second straight game with Sutter unable to join them until Monday because of NHL COVID-19 protocols (3-2 loss at the Edmonton Oilers on Saturday).
"He demands success," Hanifin said of Sutter. "Everywhere he's went, he's won. That's something we're looking forward to having in our locker room. He's going to demand a lot out of us. I'm excited to see what he brings and what he expects of us. I think it's going to be good for our team."
Gaudreau made it 3-2 at 3:27 with a power-play goal, and Hanifin scored off a centering pass from Dillon Dube at 12:17 to tie it 3-3.
"It's a tough division," Hanifin said of the Scotia North Division. "There's going to be a lot of close games going into third periods. We've got to get comfortable making plays and playing the right way from start to finish. I wish we had played like that earlier in the game, but it's good to see some resiliency and come back and get the point."