Hart was to play the 2018-19 season, his first as a professional, with the Philadelphia Flyers' American Hockey League affiliate in Lehigh Valley after he was named the Western Hockey League Player of the Year and Canadian Hockey League Goaltender of the Year. But injuries and inconsistency forced the Flyers to bring him up in mid-December, after 18 AHL games, and he wasted no time showing why he was taken in the second round (No. 48) of the 2016 NHL Draft.

Hart made 20 saves in his NHL debut a 3-2 win against the Detroit Red Wings in Philadelphia on Dec. 18, 2018. At 20 years and 127 days, he became the youngest Flyers goalie to win his NHL debut and the youngest NHL goalie to do so since Carey Price (20 years, 55 days) for the Montreal Canadiens on Oct. 10, 2007. He became the first goalie in Flyers history to win four straight starts before turning 21 during a streak of eight consecutive victories from Jan. 14-Feb. 9, 2019. His gifts, including the ability to minimize soft goals, were evident in juniors and in the AHL. He went 16-13-1 with a 2.83 goals-against average and .917 save percentage in 31 games (30 starts) and placed ninth in voting for the Calder Trophy, awarded to the NHL's top rookie.

Hart made 25 saves to become the youngest Flyers goalie with a shutout (21 years, 57 days) in a 4-0 victory against the New Jersey Devils on Oct. 9, 2019, and the youngest to win a Stanley Cup Playoff game (21 years, 355 days) with 34 saves in a 4-1 win against the Boston Bruins in the opener of the round-robin portion of the Stanley Cup Qualifiers on Aug. 2, 2020. Pete Peeters held the previous record (22 years, 235 days) when his 14 saves helped Philadelphia win Game 1 of the 1980 Preliminary Round 4-3 against the Edmonton Oilers at The Spectrum.

Hart signed a two-year, $4 million contract ($2 million average annual value) with the Vegas Golden Knights on Oct. 24, 2025. He joined the organization on a professional tryout contract Oct. 16.