Slafkovsky scored to extend Montreal's lead to 5-2 at 7:05 of the third period. Suzuki and Caufield had the assists.
It was Slafkovsky's first 5-on-5 goal of the Stanley Cup Playoffs and Caufield's second. Suzuki had two in the first two rounds.
"I thought we were good," Suzuki said. "I think 'Slaf' and Cole were using their speed, making plays. I thought we defended pretty well and we were able to transition on them quickly. It's nice to see us get going in the first game, and hopefully we can keep doing that."
The first two rounds presented challenges for Suzuki, Slafkovsky and Caufield. They saw a lot of Tampa Bay Lightning center Anthony Cirelli in the first round. Cirelli, like Suzuki, is one of three finalists for the Selke Trophy, which is awarded to the best defensive forward.
They did their best to neutralize that matchup, but it didn't lead to any offense.
They were less matchup-based in the second round against the Buffalo Sabres, though they did get on the ice a lot against Tage Thompson, which created a focus on limiting chances against, perhaps a reason why they had fewer chances for.
Suzuki, though, talked about the need for 5-on-5 production from his line on Wednesday shortly after the Canadiens arrived here.
"Five-on-5, definitely have to find a way to produce more if we're going to keep moving forward," he said. "It's not one thing that's missing, it's just little plays here and there. I think all three of us can do a better job. Definitely exciting for our group that we haven't been producing where we usually are at. Guys have stepped up and helped us out. We've got to return the favor and play a little better."
They did that in Game 1, and at key times in the game.
Caufield's goal came 27 seconds after the Hurricanes stung the Canadiens early. Slafkovsky's goal came at a time when it felt like Carolina had the ice tilted and was one goal away from turning up the heat and igniting the crowd into thinking the comeback was possible.
"They played great today," Montreal forward Jake Evans said. "They wanted to make plays. They got some good momentum swings for us back. That fifth goal was huge to kind of push them back a bit."