Antti Niemi was pulled after giving up three goals on nine shots in the first 13:57 and was replaced by Carey Price, who made 24 saves.
"Gotta be better," Domi said. "Next game is absolutely huge. We've got to start winning some hockey games here. We know what we have to do.
"We're going to stick together as a group. We've got confidence in one another. [If] we work hard and we push each other and hold each other accountable, we'll be fine."
Denis Malgin, Frank Vatrano and Riley Sheahan also scored for Florida (25-24-8), which is 8-4-0 in its past 12 games. James Reimer made 28 saves in his second consecutive start.
"I liked our start, but after the first 10 minutes we looked at the score 3-1 and thought it was going to be easy," Panthers coach Bob Boughner said. "That's a team that's fighting for their life. It was a good lesson to learn and we stayed composed the second period."
Barkov gave the Panthers a 1-0 lead at 3:59 of the first period with a power-play goal after Niemi made a save on Keith Yandle's shot from the point.
After Malgin made it 2-0 at 11:28, Domi scored his first goal at 13:04 when Tomas Tatar stripped Mike Hoffman in the Panthers zone and sent Domi on a breakaway.
Vatrano's 20th goal of the season made it 3-1 at 13:57 when his shot from close range deflected off Shea Weber's skate. Niemi was pulled for Price after the goal.
Niemi made an NHL career-high 52 saves in Montreal's 5-1 win against Florida on Jan. 15 at Bell Centre.
"It's one of those nights where it happens to every goalie," Canadiens coach Claude Julien said. "It's tough. It was tough on him tonight early in the game. Sometimes those things just happen.
"It's my job to make a decision, and I thought that by bringing (Price) back in, it would give us an opportunity to get back in the game, which we did. But we didn't finish what we started."
Gallagher made it 3-2 at 16:46 with a wrist shot off a 2-on-1 before Domi tied the score 3-3 at 5:18 of the second period when he redirected Mike Reilly's pass.
Barkov made it 5-3 at 9:13 of the third period to match his NHL career high with four points.
"Every team's got those guys," Canadiens defenseman Jordie Benn said of Barkov. "He's their team's special player. He's a big boy that can put the puck in the net. He had a pretty good night. We got the short end of the stick."
Sheahan scored an empty-net goal with 2:25 left for the 6-3 final.
