Team USA led 2-0 in the third period when Slovenia rallied to create a tense finish.
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. But a goal by Mursak, who slipped the puck slip a puck past US goaltender Ryan Zapolski with one minute, 37 seconds left to tie the game and force overtime.
Tony Granato, a former assistant coach of the Red Wings, who is how head coach at Wisconsin, and Chris Chelios, who is now an assistant coach in Detroit, are two of the United States coaches.
But it was a day for Mursak and Slovenia.
Three-time Olympian Jocelyne Lamoureux-Davidson was huge for the USA against the Olympic Athletes of Russian Federation en route to creating a piece of history. After being blanked in its first two games of the preliminary round, Griffin's accomplishment will prove to be a building block for the Korean hockey program. He finished his Red Wings career with four points in 46 games before moving to the KHL.More news: Carlos Carvalhal: Swansea put 'all the meat in the barbecue', says manager
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.
Everything eventually collapsed on Team USA when Mursak scored the tying tally with less than two minutes to play with the goaltender pulled for Slovenia.
Blaz Gregorc, who had a goal and an assist for Slovenia, fired shot on goal from the point. He was on the ice for Slovenia's first goal, but not the final two.
"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.
This time, the pass made it through, and Mursak beat Zapolski on the stick side, to win the game. It would turn out to be the game-winning goal, sending Slovakia to the top of Group B.