Lagace played four seasons in the Quebec Major Junior Hockey League, the last two after signing a three-year, entry-level contract with the Dallas Stars on July 23, 2012. He spent three seasons in the Stars organization, bouncing between the American Hockey League and the ECHL.
The Vegas Golden Knights signed him as a free agent on July 1, 2017. He was assigned to Chicago of the AHL but was recalled in late October after the Golden Knights lost starting goaltender Marc-Andre Fleury and backup Malcolm Subban to injuries. Lagace made his NHL debut against the New York Islanders on Oct. 30, 2017, after another goaltender, Oscar Dansk, left with an injury.