Brian O'Neill scored in the first period and Jordan Greenway added a goal in the second to put the U.S.in front 2-0.
It is the first Winter Olympic men's hockey tournament in 24 years without NHL players.
With under two minutes left in the third, Slovenia pulled its goalie, Gasper Kroselj, and Mursak scored to tie the game at 2.
The Olympic athletes from Russian Federation team also suffered a surprise opening defeat as Slovakia's Peter Ceresnak scored the decisive goal in their 3-2 win.
Slovenia took just three shots in the first period but continued to create more chances for itself as the game progressed. Next up for Team USA is a faceoff with rival Canada, which won the gold in the 2014 Sochi Games. 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. Ceresnak's slap shot from the point found twine to capitalize on a power play and give his team a third-period lead against the gold-medal favorites. The U.S. netminder had very little chance to stop either deciding goal. In the end, Canada defeated Sweden, 3-1, in the final to take home gold, while the United States defeated the Czech Republic, 9-3, in the bronze medal game.
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"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.
In overtime, Mursak was left alone in front of the U.S.net to score the game-winning goal off a pass from Gregorc.
Slovenia, the Americans' first opponent in the round-robin preliminary round, enters with much lower expectations.
Of course, it's an upset for the U.S. Even without National Hockey League players on the team they were expected to beat Slovenia.
Russia, by the way, might have not only lose the game but a player for at least a game - Per IIHF source Nikita Nesterov's hit in the 1st period will be reviewed by disciplinary committee.
In the United States, the game starts Wednesday morning at 7:10 a.m. ET and will be broadcast on NBC Sports Network.