But things changed in the third period.
Jan Mursak scored the victor for Slovenia 38 seconds into the overtime period. Kroselj stopped 34 of the 36 shots he faced to pick up the victory, giving up goals to Brian O'Neill and Jordan Greenway. It would turn out to be the game-winning goal, sending Slovakia to the top of Group B.
In the second period Jordan Greenway, the first African American to play for the US men's Olympic hockey team, tapped in a puck that popped loose from a scrum in front of Kroselj.
Slovenia, ranked 15th by the International Ice Hockey Federation, got on the board in the third period when Blaz Gregorc threaded a wrist shot through a crowd in front of USA goalkeeper Ryan Zapolski.
With some pressure on, after a penalty to US captain Brian Gionta, Slovenia pulled their goalie for an extra attacker. With under two minutes left in the third, Slovenia pulled its goalie, Gasper Kroselj, and Mursak scored to tie the game at 2.
Blaz Gregorc, who had a goal and an assist for Slovenia, fired shot on goal from the point.
Slovenia, the Americans' first opponent in the round-robin preliminary round, enters with much lower expectations.
The big center wheeled to his left, pirouetted, carried the puck along with him through the rotation and fired it cleanly behind Zapolski.
Which in all likelihood makes Thursday's (Wednesday night on the West Coast) first-round game between the sport's two superpowers at Kwandong Hockey Centre a sneak preview of the Games' gold medal final next Thursday.
"Hopefully, we can start looking to the next game and surprise somebody else".
"I'm proud of my guys, I'm proud of the team", said Mursak, who played sporadically with the NHL's Detroit Red Wings between 2010 and 2012.
It was the first win for Slovenia over the United States in Olympic men's ice hockey.