But a goal by Mursak, who slipped the puck slip a puck past USA goaltender Ryan Zapolski with one minute, 37 seconds left to tie the game and force overtime.
Brian O'Neill and Jordan Greenway scored as the U.S. built a 2-0 lead.
For the United States, the loss represented a huge stumble out of the gate in the first quadrennial Olympic tournament played without National Hockey League players since 1994.
Kroselj made a handful of quality saves to keep Slovenia in the game, and Zapolski had to do the same in the second period, stopping Urbas on a one-timer with Greenway in the box and the US on the penalty kill.
In order to achieve the best possible finish in Group B, the United States must knock off Slovenia and Slovakia and hope to play the Olympic Athletes from Russian Federation to a tight game.
But things changed in the third period.
Team USA's debut in the 2018 Winter Olympics degenerated into a nightmare.
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USA coach Tony Granato showed a lot of trust in Greenway by having him center the top line with Ryan Stoa and captain Brian Gionta on his wings. All four have significant worldwide experience, but Terry is probably the most recognizable name as he was the hero who scored the shootout-winning goal in the gold medal game at last year's World Junior Championships. Slovenia only gets two, instead of three, since the win did not come in regulation.
Blaz Gregorc, who had a goal and an assist for Slovenia, fired shot on goal from the point. Canada walked away with a gold medal after the 2014 Olympics, but Team USA will look to rebound as it enters the Winter Games as co-favorites for the gold (live starting at 10:10 p.m., NBCSN).
This time, the pass made it through, and Mursak beat Zapolski on the stick side, to win the game.
Returning to the lineup later in the season, he had trouble finding any playing time, the Red Wings put him on waivers and Mursak signed with Amur Khabarovsk of the KHL.
It was the first win for Slovenia over the United States in Olympic men's ice hockey. The second game for the USA was to be played against Canada, and no one knew what to expect.
US will face Slovakia next, at 10:10 p.m. ET on February 16th.