Hart was considered one of the top goaltender prospects in the 2016 NHL Entry Draft. NHL Central Scouting named him the top available North American goaltender in their 2016 midterm rankings. He was named the CHL Goaltender of the Year for 2016 on May 28, 2016. He finished the year as the second-ranked North American goaltender in the NHL Central Scouting final rankings.

Hart was drafted 48th overall in the 2016 NHL Entry Draft by the Philadelphia Flyers, the first goaltender selected in 2016. He signed an entry-level contract with the Flyers on October 2, 2016.

Hart was named both the WHL Player of the Year and Goalie of the Year for the 2017–18 WHL season after he compiled a 1.60 GAA, a 0.947 save percentage and seven shutouts. In addition to being named WHL Goalie of the Year for the 2017–18 season, he was named CHL Goaltender of the Year for the second time, the first being in 2016. He was the first goaltender in CHL history to win the award twice.

Hart began the 2018–19 season with the Flyers American Hockey League affiliate, the Lehigh Valley Phantoms. He was recalled on December 17, 2018, after goaltender Anthony Stolarz was placed on injured reserve. Hart made his NHL debut the following day in a 3–2 win over the Detroit Red Wings and was named first star of the game. He became the youngest goaltender since Carey Price to win in his NHL debut and the youngest ever in Flyers history. After recording his fourth straight win as an NHL starter against the Winnipeg Jets on January 28, Hart became the first goaltender since Steve Mason to record four straight wins as a starter before the age of 21. Hart was named NHL Rookie of the Month for January 2019, going 6–2–1 with a 2.33 goals-against average and a .931 save percentage. He was later named the NHL Second Star of the Week for the week of February 4 after going 3–0–0 and extending his win streak to six. Hart was also one of eight goaltenders Philadelphia used during the season, setting a new record for the NHL.

On April 29, 2019, he was officially named to the Team Canada roster for the 2019 IIHF World Championship. As the backup goaltender to Matt Murray, Hart collected 3 wins in as many games to help Canada progress through to the playoff rounds before losing the final to Finland to finish with the Silver Medal on May 26, 2019.

Hart attended the Flyers training camp prior to the 2019–20 season. He entered the preseason weighing 191 pounds, nearly 10 more than the previous season. On October 9, after defeating the New Jersey Devils by a score of 4–0 in Philadelphia’s home opener, Hart became the youngest goaltender in Flyers franchise history to record a shutout, doing so at the age of 21 years, 57 days. After suffering a lower right abdominal muscle strain during practice on January 15, 2020, Hart missed nine consecutive games before returning on February 10 in a 4–1 win over the Florida Panthers. Hart made his postseason debut on August 2 winning a round robin game against the Presidents’ Trophy winning Boston Bruins 4–1. Hart made 34 saves in the win and became the youngest Flyers goaltender to win a game in the postseason. On August 16 and 18, Hart posted consecutive shutouts, making him the youngest Flyers goaltender to record a shutout and the second-youngest in NHL history to consecutive back-to-back playoff shutouts.