The Flames trailed 2-0 before scoring four straight goals.
Lucic scored his 200th NHL goal on the power play to cut the lead to 2-1 at 18:14 of the first period. He has scored in consecutive games after going the first 27 games of the season without one.
"I think it's pretty cool, too, I scored my first goal and my 200th goal against the same team," Lucic said. "Nice to finally hit that milestone."
Rinaldo tied it 2-2 at 3:01 of the second period with his first goal in 20 games dating to Oct. 19, 2018 as a member of the Nashville Predators, and Johnny Gaudreau set up Monahan after an end-to-end rush to put Calgary up 3-2 at 11:46.
Dube scored on a rebound 1:29 into the third period to extend the lead to 4-2.
"When a young guy comes in and provides what he's been providing it sometimes re-energizes us older guys to play the way that we can," Lucic said of the 21-year-old Dube. "He's been huge for me."
Matt Roy made it 4-3 at 5:24 with a point shot through traffic.
Los Angeles, which had one power-play goal in the previous seven games, scored twice with the man-advantage in the first period.
Kopitar made it 1-0 at 11:09 with a shot short-side through a screen, and Doughty scored his first in 11 games from the high slot over Talbot's glove at 14:43 to push the lead to 2-0.
"This year we've played very well with the lead," Doughty said. "So after the first period, with that good start, we felt pretty comfortable in here, maybe too comfortable. We didn't push enough in the second. Maybe sat back a little bit too much and we allowed them back in the game. Then we were chasing after we got down 3-2, obviously."
