The Rouyn-Noranda Huskies selected him in the 1st round (11th overall) of the 2011 QMJHL Import draft, from the National League B (Switzerland's second-tier pro league) team GCK Lions. In the 2010–11 season, while playing for the Lions, he was selected to represent Switzerland at the 2011 World Junior tournament. Playing 36 games for GCK, he posted 23 points (11 goals and 12 assists) as well as 20 penalty minutes. Loaned to eventual NLB champion EHC Visp, he was scoreless in two regular-season games, adding one goal and two assists plus 12 penalty minutes in 17 playoff games.
For the 2011–12 QMJHL season, Andrighetto made the switch from Switzerland and the NLB to Canada and the QMJHL. Despite his small size, Andrighetto managed to put up a total of 74 points (36 goals plus 38 assists), along with 50 penalty minutes. He was a -23 in plus/minus despite his high scoring. The Huskies made it to the playoffs but lost to the Shawinigan Cataractes in a four-game sweep. Andrighetto had only two assists and four penalty minutes in those four games.
Andrighetto returned to Rouyn-Noranda and the Huskies for the 2012–13 QMJHL season. He had a successful season in his second year playing North American hockey, playing in 53 games for the team but recording 98 points (31 goals and 67 assists) to finish sixth in regular season scoring for the league, with 45 penalty minutes as well as a +25 plus/minus rating. With the Huskies once again in the playoffs, they made a run to the semi-finals, falling to the Halifax Mooseheads. Andrighetto played in all of 14 games, with 8 goals and 22 assists for 30 points with only 14 penalty minutes.
Andrighetto was entering his final year of eligibility for the 2013 NHL Entry Draft; he was 20 years of age when the draft came, held in Newark, New Jersey. The Montreal Canadiens owned the 86th pick in the third round of the draft and made him the second Swiss player selected (and first forward hailing from the country), behind Mirco Mueller, who the San Jose Sharks took at pick number 18.
The Canadiens signed him to his three-year, entry-level contract on 15 July 2013, detailed to a two-way deal, worth $625,000 if Andrighetto played in the NHL and $70,000 if he played in the AHL. The contract also included a $75,000 signing bonus.
Andrighetto reported to the Hamilton Bulldogs for the 2013–14 season, which would be his rookie campaign in the AHL. He garnered 44 points (17 goals and 27 assists) in 64 games played and added 40 penalty minutes with a -4 plus-minus rating.
Andrighetto returned to the Bulldogs for the 2014–15 season, not able to make the Canadiens out of training camp and preseason. He scored three points in a game on 2 December 2014 (against the Adirondack Flames at the Glens Falls Civic Center), with a goal and two assists. He was recalled by the Canadiens in December and made his NHL debut on 6 December against the Dallas Stars, recording his first career goal. It was the Canadiens' only goal in the 4-1 loss to the Stars, assisted by fellow European forwards Tomas Plekanec and Jiri Sekac. He registered a point in his first three games to become the first Canadiens to do so since Pierre Mondou in 1978.
In the 2015–16 season, Andrighetto was assigned to begin the year with the Canadiens new AHL affiliate, the St. John's IceCaps. He was recalled to Montreal after producing 23 points in 26 games on November 20, 2015. In remaining with the Canadiens for the remainder of the season, Andrighetto's versatility was used throughout the lineup in scoring a career best, 7 goals and 17 points in 40 games.
With the expectation of establishing a roster spot for the following 2016–17 season, Andrighetto was slow to start at training camp and was surprisingly re-assigned to start the year with the IceCaps. Scoring a more than a point-per-game pace in the AHL, Andrighetto earned a recall to join Montreal.
In struggling to secure a defined role in the team and having contributed with 8 points in 27 games, Andrighetto was dealt at the NHL tradeline by the Canadiens to the Colorado Avalanche in exchange for Andreas Martinsen on March 1, 2017. He made his Avalanche debut at the Pepsi Center in a 3-0 shutout defeat to the St. Louis Blues on March 5, 2017. As the first Swiss skater to appear for the franchise, Andrighetto recorded his first point with the Avalanche on an assist, in a 3-1 victory over the Carolina Hurricanes in his second game. On March 11, 2017, he notched his first goal for the Avalanche in a 4-2 defeat to the Ottawa Senators. Given an extended opportunity in a scoring role on a line with Nathan MacKinnon and Mikko Rantanen, Andrighetto blossomed to lead the Avalanche in scoring through the month of March and finish the season with 16 points in 19 games.
After he was protected in the 2017 NHL Expansion Draft, Andrighetto later agreed to a two-year, $2.8 million contract extension with Colorado on June 28, 2017.