With Jimmy Howard placed on injured reserve Saturday with a groin injury, Mrazek started for the second time in nine games.
"Every minute on the ice gives me more confidence and gets me ready for the next game," said Mrazek, who began the season as Detroit's No. 1 goalie.
Montreal started a 12-day, five-game road trip.
"This was a big win for sure, because you always want to get two points at the start of a trip," Galchenyuk said.
Detroit went to overtime for the third time in four days, including a 5-4 victory against the New Jersey Devils on Friday and a 2-1 shootout win at the Buffalo Sabres on Wednesday.
Detroit had 15 shots and two power plays in the second period but couldn't beat Price until the third, when Justin Abdelkader put a wrist shot over the goalie's catching glove at 3:20 for a 1-0 lead. Frans Nielsen set up the goal for his fourth point in two games.