Undrafted in the NHL, due to his diminutive stature, Agozzino was signed upon the completion of the IceDogs season to an amateur try-out contract with the Peoria Rivermen of the American Hockey League (AHL), an affiliate of the Blues.

For a third consecutive season, he led the team with 40 goals and produced a career high 48 assists and 88 points, to finish fifth in the OHL scoring race.

[8] In his Monsters debut on opening night, Agozzino scored his first professional goal and game-winner in a 2–1 victory over the Oklahoma City Barons on October 12, 2012.

[9] After initially claiming a role on the checking lines, he quickly and unexpectedly established himself amongst the Monsters scoring forwards to begin the season.

[12] On March 22, 2013, Agozzino's smooth transition to the professional ranks was completed when he was signed to a two-year entry-level contract with the Colorado Avalanche, the NHL parent club of the Monsters.

[13] With Lake Erie out of the playoffs, he finished the regular season as one of only two skaters to have played in every game for the Monsters, whilst leading the team with 32 assists and 52 points.

[17] Having left the Avalanche organization as a free agent, Agozzino agreed to a one-year, two-way contract with the St. Louis Blues on July 1, 2016.

Following his adjustment to a new playing system, Agozzino remained in the AHL for the duration of the season in continuing his career scoring pace with 18 goals and 54 points in 71 games.

[19] A year later, after leaving St. Louis as a free agent, Agozzino opted to return to his original club, the Colorado Avalanche, in signing a two-year, two-way contract.

[25] In the offseason, having left the Senators as a free agent, Agozzino was signed to a two-year, two-way contract with the San Jose Sharks on July 13, 2022.

[26] On June 27, 2023, Agozzino was traded by the Sharks in a return to the Anaheim Ducks organization in exchange for pending free agent Andrej Šustr.