After Smith scored for Nashville 54 seconds into the third to tie the game 2-2, Hathaway gave Calgary a 3-2 lead at 6:03 when he deflected Matthew Tkachuk's point shot.
"We ran a play," said Kylington, who has played seven NHL games, six this season. "[Tkachuk] said he wanted me to take the puck down the wall and then he wanted to pop up in the middle, and we won the puck clean off the face-off. Then I did what he said and I passed the puck, and [Hathaway] made a great tip. It was a good play."
Alan Quine, recalled from Stockton of the American Hockey League on Friday and playing his first game for the Flames, scored on a backhand deke to make it 4-2 at 12:12. It was Quine's first goal in 45 games, dating to Jan. 31, 2017, when he played for the New York Islanders.
"Obviously, they've had a great start to the season," Quine said of the Flames. "To be able to fit in and contribute, it's a great feeling. You just want to keep on winning. That's the ultimate goal."
Lindholm scored into an empty net with three minutes left for the 5-2 final score. He has 11 points (five goals, six assists) in a six-game point streak, and his 16 goals are one shy of his NHL career high, set in 2014-15 with the Carolina Hurricanes.
"I thought we just gave them too many," Craig Smith said. "[Saros] is a good goalie. We gave them too much. I thought they capitalized on a lot of quick turnovers. We're going to have to go back and look at the tape and make some adjustments again and correct it."
Kylington scored from the slot on Calgary's first shot of the game 1:12 into the first period to put Calgary up 1-0.
"Amazing feeling," Kylington said. "First time I touched the puck there. When I got it off my stick, I had a good feeling, and when I saw the puck in the back of the net, happy feeling."
Sissons redirected Roman Josi's pass to tie it 1-1 at 19:07.
Monahan scored on his backhand from the top of the crease at 16:40 of the second period to give Calgary a 2-1 lead. He has nine points (five goals, four assists) in the past five games.