In 2016, Murray won his first Stanley Cup championship as the starting goaltender for the Penguins during the playoffs after having replaced then-starter Marc-André Fleury.

He spent his younger years playing AA before moving up to the Thunder Bay Kings AAA program.

Murray led the Penguins past the San Jose Sharks in the finals to the franchise's fourth Stanley Cup.

[10] He posted an impressive .923 save percentage and 2.08 GAA throughout the playoffs and became only the sixth starting rookie goaltender in the Stanley Cup Finals since 1976.

[12] During warm-ups of game one of the Penguins' first-round matchup versus the Columbus Blue Jackets in the 2017 Stanley Cup playoffs, Murray was injured and Fleury replaced him.

[13] A thrilling game seven overtime win helped Murray and the Penguins reach the Stanley Cup Finals for the second year in a row.

[16] Murray made 27 saves in game six, shutting out the Predators 2–0 and winning the Stanley Cup for the second consecutive season.

On February 26, 2018, during practice, Murray was struck in the head with a puck shot by teammate and defenseman Olli Määttä and was forced to leave the session early.

[21] On April 23, 2018, Murray was nominated for the King Clancy Memorial Trophy as a player who best exemplifies leadership qualities and gives back to in his community.

[23] He returned to the lineup on October 13 as a backup to Casey DeSmith[24] only to be placed on injured reserve again in November due to a lower-body injury.

On October 12, 2019, Murray recorded his 100th NHL win, a 7–4 victory over the Minnesota Wild; he accomplished the feat in only 166 games, tying Pete Peeters for seventh fastest since the 1967 expansion.

[26] On October 7, 2020, as an impending restricted free agent, Murray was traded by the Penguins to the Ottawa Senators in exchange for Jonathan Gruden and a second-round pick in 2020.

On November 28, 2021, Murray was placed on waivers and later demoted to regain his game with AHL affiliate, the Belleville Senators, the following day after going unclaimed.

[29] After two games in Belleville, managing a 1–1–0 record and a .918 save percentage, Murray was recalled to Ottawa to resume duties as the starter.

[32][31] Shortly after the vetoed trade to the Sabres, rumors began to publicly appear heavily connecting Murray to interest from the rival Toronto Maple Leafs.

[47] On April 29, 2019, Murray was named as the first choice goaltender to backstop Canada at the 2019 IIHF World Championship held in Slovakia.

[48] Recording five wins in seven games, Murray helped Canada progress through to the playoff rounds before losing the final to Finland to finish with the silver medal on May 26, 2019.