Game 2 in the best-of-three series is Thursday night.
Patrice Bergeron's goal midway through the third period created a two-goal cushion.
Carey Price finished with 32 saves for the Canadians, who have won two straight Olympic gold medals and 15 consecutive best-on-best hockey games since losing to the U.S. at the 2010 Vancouver Games.
Slovakia's Tomas Tatar scored his third goal in two games 7 minutes into the second period to get Europe within one.
Jaroslav Halak, who is also from Slovakia, stopped 35 shots for the unique team made up of eight European nations outside of the continent's traditional powers.
The Canadians were playing at home, but didn't seem to get much of an emotional boost in the Toronto Maple Leafs' arena, where there were a lot of empty seats and suites.
The NHL and NHL Players' Association combined to revive the event, which hasn't been held since 2004, with a twist. They created Team Europe and Team North America, a squad made up of players 23 and younger from the U.S. and Canada.
The format created a highly competitive eight-team field, but as Canada coach Mike Babcock said earlier in the day, "The World Cup is great. It's not the Olympics. Let's not get confused."
There was no confusion about who had more talent on the ice Tuesday night, but Europe got off to a strong start and put pressure on a star-studded squad that has been behind just twice in five games and for a little more than a minute each time against the U.S. and Russia.
Price had to make three saves and face a power play in the first 21 seconds of the game.
Shortly after killing off the penalty, Bergeron carried the puck up the middle of the ice, faded to the right and passed across the crease to set up Marchand's fourth goal of the tournament 2:33 into the game. The Boston Bruins teammates have been playing on a line with Pittsburgh Penguins superstar Sidney Crosby, and the trio rank among the top three in World Cup scoring.
Another Bruin, Zdeno Chara, was careless with the puck later in the opening period and it was costly. Ryan Getzlaf snatched the puck from Chara near center ice along the boards and faked a shot at the top of the left circle, leading to a tap-in goal for Stamkos on the right side of the net.
Crosby's second assist of the night and his World Cup-high ninth point helped Canada to a 3-1 lead.
He shot the puck on the left side of the net, got to the carom and circled back to set up Bergeron in front of the net.
The Canadians didn't relax with the lead, attacking on offense and backchecking on defense to prevent Europe from having hope of coming back even when it pulled Halak late in the game.
Team Europe was without one of its key players, Marian Gaborik, who suffered a foot injury in a semifinal win over Sweden that will keep him out of the Los Angeles Kings' lineup for eight weeks. Gaborik was replaced in the lineup by Mikkel Boedker, the only skater on the ice making his first appearance in the two-week tournament.