Joseph was selected by the Saint John Sea Dogs in the 2013 Quebec Major Junior Hockey League (QMJHL) entry draft in the third round, 51st overall.

[8] During the 2016–17 playoffs, Joseph was named Canadian Hockey League Player of the Week after scoring two goals and three assists in two games.

Joseph went on to help the Sea Dogs capture the President's Cup as QMJHL champion over the Blainville-Boisbriand Armada in a four-game sweep.

[11] Joseph began his professional career in the 2017–18 season with the Tampa Bay Lightning's American Hockey League (AHL) affiliate, the Syracuse Crunch.

[13] At the conclusion of the Lightning's training camp ahead of the 2018–19 season, Joseph found out he had made the opening night roster from general manager Julien BriseBois—before boarding the team's charter plane after a 3–2 Lightning win against the Florida Panthers, BriseBois informed Joseph he had made the roster.

[18][19] On November 4, Joseph scored his first NHL goal in a 4–3 Lightning overtime win over the Ottawa Senators away at the Canadian Tire Centre.

[21] Following the murder of George Floyd and shooting of Jacob Blake, Joseph worked alongside goaltender Curtis McElhinney in creating a mask to represent social justice.

[23] The following season, Joseph played a larger role with the Lightning and won a second Stanley Cup defeating his hometown Montreal Canadiens in the final.

[26] On April 1, Joseph registered his first hat trick versus goaltender Alex Nedeljkovic in a 5–2 win over the Detroit Red Wings.

[28] As a restricted free agent in the following off-season, Joseph was re-signed by the Senators to a four-year, $11.8 million contract extension on July 29.

[36][37] On April 29, 2019, Joseph was named to make his senior debut for Team Canada at the 2019 IIHF World Championship held in Slovakia.