Elias Pettersson had two goals and an assist, Brandon Sutter and Bo Horvat scored, and Jacob Markstrom made 30 saves for the Canucks.
"It's frustrating standing here with a loss," Markstrom said. "We took some penalties and they kept going. We have to learn as a team, especially up by one coming into the third, it's not easy to do on the road. You can't just waste those."
Monahan tied it 4-4 with a power-play goal at 9:27 of the third period on a backdoor pass from Gaudreau, and Lindholm, acquired with defenseman Noah Hanifin in a trade with Carolina on June 23, scored his second, also on the power play, at 14:28 to give the Flames a 5-4 lead.
"I just tried to shoot it as fast as possible," Lindholm said. "It probably wasn't my best shot but it went in. It was a really good pass and a nice play by the [power play] there. It's good to see it go in and nice to get the win."
Czarnik scored into an empty net at 18:22 for a 6-4 lead, and Frolik scored a shorthanded empty-net goal with 36 seconds remaining to make it 7-4.
"Up a goal, down a goal, up a goal," Gaudreau said. "It was a lot of resilience there in the third. I was pretty happy with the way our team played."
Lindholm gave the Flames a 1-0 lead 12 seconds into the first period. Pettersson tied it 1-1 at 13:15 to become the sixth player in Canucks history to score at least one goal in each of his first two games, and the first since Rick Lanz in October 1980.
"Personally, I'm very happy with my start," Pettersson said. "It's going good out there. I'm confident and producing offensively and I think I'm playing good defensively, too."
Sutter scored shorthanded at 16:38 give Vancouver a 2-1 lead before Giordano scored 30 seconds later to make it 2-2.
Pettersson, 19, the No. 5 pick in the 2017 NHL Draft, scored on a one-timer on the power play 2:52 into the second period to give the Canucks a 3-2 lead, but Gaudreau scored on a rebound on the power play at 9:31 to tie it 3-3.
"We were moving the puck around pretty well tonight," Gaudreau said. "Lots of practice the past few days, a lot of video on the power play. I think we definitely got a little bit better there. Still some room for improvement."
Pettersson's centering pass on a 5-on-3 power play was redirected by Horvat at 16:58 to give the Canucks a 4-3 lead.