Sgarbossa first played midget hockey with the Halton Hurricanes of the South-Central Triple A Hockey League before he was selected 54th overall in the 2008 OHL Priority Selection by the Barrie Colts. In his first Major junior season with the Colts he finished fourth in team scoring with 43 points in 67 games. Adapting to the OHL, Sgarbossa increased his scoring rate and was traded midway through the 2009–10 season, to the Saginaw Spirit.
Sgarbossa was passed over in the 2010 NHL Entry Draft but was signed as a free agent by the San Jose Sharks to a three-year entry level contract on September 21, 2010. Sgarbossa was signed after appearing in the NHL YoungStars tournament for the Sharks, scoring 3 goals and 4 points in 2 contests. Sgarbossa was again on the move in the OHL the following 2010–11 season when he was traded by the Spirit to the Sudbury Wolves. He immediately made an impact with the Wolves to lead the team in scoring for the remainder of the campaign with 62 points in only 37 games.
During the 2011–12 season, whilst with the Wolves, Sgarbossa's NHL rights were traded by San Jose at the trade deadline on February 27, 2012, to the Colorado Avalanche, along with Jamie McGinn and Mike Connolly in exchange for Daniel Winnik, TJ Galiardi and a seventh round draft selection. At the conclusion of the season, marking the end of his junior career, Sgarbossa became the first Sudbury Wolves player in 33 years, since Mike Foligno, to claim the Eddie Powers Memorial Trophy as the OHL's leading point scorer with 102 points. Finishing second in league MVP voting and earning a selection in the First All-Star team, Sgarbossa was given the accolade as Sudbury Professional Athlete of the year.
In his first professional season in 2012–13, Sgarbossa was directly assigned by the Avalanche to their AHL affiliate, the Lake Erie Monsters, due to the ongoing NHL lockout. He made his professional debut on opening night in a 2–1 victory over the Oklahoma City Barons on October 12, 2012. Sgarbossa quickly established himself as one of the Avalanche's top prospects by immediately becoming an offensive leader for the Monsters. At the midpoint of the season, and leading the Monsters in scoring, tied for second amongst rookies in the AHL, he was selected alongside fellow Monsters rookie Andrew Agozzino to participate in the AHL All-Star Game. Straight after contributing to a Western Conference All-Star victory, Sgarbossa received his first NHL recall to the Avalanche on January 29, 2013. He made his NHL debut the next day for the Avalanche in a 3–0 road defeat against the Vancouver Canucks. After six scoreless games with Colorado, Sgarbossa was returned to Lake Erie, to continue his season. On February 26, his ability to also play with an edge was evident as he was suspended by the AHL for one game after an slashing incident with Michael Liambas of the Milwaukee Admirals.
In the 2014–15 season at the trade deadline, Sgarbossa was traded by the Avalanche to the Anaheim Ducks in exchange for Mat Clark on March 2, 2015. He was assigned to AHL affiliate the Norfolk Admirals.
Sgarbossa began the 2016–17 season within the AHL with the San Diego Gulls before he was recalled to the Ducks. He had appeared in a career high 9 games with 2 assists for the Ducks before he was dealt to the Florida Panthers in exchange for Logan Shaw on November 16, 2016. He was instantly re-assigned to the Panthers affiliate, the Springfield Thunderbirds. Sgarbossa impressed with the Thunderbirds and was quickly recalled to the Panthers. Enjoying his longest stint in the NHL, Sgarbossa closed out the season in Florida, appearing in 29 games for 2 goals and 7 points.
Following the completion of the season with the Panthers, Sgarbossa was not tendered a qualifying offer as a restricted free agent. On July 1, 2017, Sgarbossa signed a one-year, one-way $650,000 deal with the Winnipeg Jets.