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The Olympic scoring leader four years ago, Slafkovsky, found the back of the net in the first and third periods, while forward Dalibor Dvorsky scored in the second and centre Adam Ruzicka sent one into the empty net in the Group B clash.
Finland outshot Slovakia 40-25 but were left frustrated by missed chances as goalie Samuel Hlavaj turned away 39 shots in a chippy affair with NHL players back in the Olympics for the first time since 2014.
Both teams next play on Friday, with Slovakia taking on hosts Italy as Finland face Nordic rivals Sweden.
"It wasn't about the one, two, or three players; it was about the 22 guys who battled today," said Slovakia coach Vladimir Orszagh.
"You know, there were some mistakes, of course. We got a great goalie who helped us to keep us in the game, but... it was a great team effort."
Fans wearing NHL jerseys crowded in with the flag-waving fans, who cheered wildly as the Montreal Canadiens' 21-year-old winger Slafkovsky got Slovakia on the board with a fine solo effort that ricocheted in off the post in the eighth minute.
A skirmish near the Slovak net ahead of the intermission saw Finland forward Joel Armia in the penalty box for the first two minutes of the second period on a roughing penalty, but the Slovaks could do nothing on the power play.
Once out of the box, Armia fed the puck perfectly to Eeli Tolvanen, and the Seattle Kraken winger whipped it past the Slovak goalie to level the contest.
Old NHL grudges spilt onto the Olympic ice as Finnish defenceman Niko Mikkola, who plays for the Florida Panthers, got into it with the Tampa Bay Lightning's Erik Cernak at the end of the second, but both sides regained their composure.
Dvorsky put Slovakia ahead when he flipped the puck over Finnish alternate captain Sebastian Aho, who had fallen onto the ice in front of the net.
Slafkovsky added an insurance goal on a power play, and Ruzicka scored into the empty net after Finland pulled their goalie for an extra skater late in the third.
