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Domingue was drafted 13th overall by the Moncton Wildcats in the 2008 Quebec Major Junior Hockey League (QMJHL) Entry Draft, and he played major junior hockey in the QMJHL with both the Wildcats and the Quebec Remparts. Then he was selected to play in the 2010 CHL/NHL Top Prospects Game. Domingue was subsequently drafted by the NHL's Phoenix Coyotes and, on June 1, 2011, he was signed by the team to a three-year, entry-level contract.
In the 2014–15 season, after Arizona traded its backup goaltender Devan Dubnyk to the Minnesota Wild, Domingue was called up to the NHL for the first time to back-up starter Mike Smith. Domingue made his NHL debut appearing in relief and allowing two goals in a 7–2 defeat to the Ottawa Senators on January 31, 2015. The following day, Domingue received his first NHL start, and recorded his first win, in a 3–2 victory over the then-Eastern Conference-leading Montreal Canadiens. He finished the season as Smith's full-time backup.
In the 2015–16 season, Domingue was again recalled, following an injury to Mike Smith. Originally intended to back-up Anders Lindbäck during Smith's injury, Domingue's strong play moved him ahead of Lindbäck and into the temporary starting role. On June 27, 2016, Domingue signed a two-year contract extension with Arizona.
Domingue began the 2017–18 season as Arizona's backup goaltender. With an injury to newly-acquired starter Antti Raanta from the New York Rangers, Domingue was unable to hold the fort, going winless in seven games. On October 29, 2017, Domingue was placed on waivers by the Coyotes, but cleared waivers the next day, and was the assigned to the Coyotes' AHL affiliate, the Tucson Roadrunners, on October 31, 2017. On November 14, 2017, before appearing with the Roadrunners, the Coyotes traded Domingue to the Tampa Bay Lightning in exchange for goaltender Michael Leighton and forward Tye McGinn.
On June 22, 2018, the Lightning signed Domingue to a two-year, $2.3 million contract extension. On December 10, 2018, Domingue was named the NHL's second star of the week. On February 2, 2019, Domingue helped the Lightning record a 3–2 win over the New York Rangers at Madison Square Garden. The win was Domingue's ninth straight win. The win tied him with John Grahame (2005–06) and Andrei Vasilevskiy (2017–18) for the longest win streak by a Lightning goaltender. On February 10, 2019, Domingue set the franchise record with his 10th straight win, which came in a 5–2 Lightning victory over the Florida Panthers at BB&T Center. On February 19, 2019, Domingue extended his win streak to 11-games in a 5–2 win over the Philadelphia Flyers at Wells Fargo Center. Domingue's streak came to an end against the Boston Bruins on February 28, 2019.
On September 20, 2019, Domingue was placed on waivers by the Lightning.He cleared the following day, and was assigned to the Syracuse Crunch to begin the 2019–20 season. In four games with the Crunch, Domingue posted a 2–1–1 record.
On November 1, 2019, Domingue was traded to the New Jersey Devils in exchange for a conditional 7th round draft pick in the 2021 NHL Entry Draft. He initially joined AHL affiliate, the Binghamton Devils, before he was recalled by the Devils to serve as backup to Mackenzie Blackwood. Domingue struggled to assert himself as the Devils backup, collecting just 3 wins in 16 appearances, before he was placed on waivers and re-assigned to the AHL with Binghamton.
On February 24, 2020, Domingue was traded to the Vancouver Canucks in exchange for Zane McIntyre. He immediately joined the club due to an injury to Jacob Markström. He made a lone appearance with the Canucks in a 5-3 defeat to the Columbus Blue Jackets on March 1, 2020. Remaining on the club after the COVID-19 pandemic pause, as the Canucks' third choice goaltender in the return-to-play post-season.
As a free agent from the Canucks, Domingue joined fellow Western Canadian club, the Calgary Flames, on a one-year, two-way contract on October 11, 2020.