Agozzino made his major junior hockey debut with Niagara of the Ontario Hockey League after he was chosen by Mississauga with the No. 15 pick in the 2007 OHL Draft before the IceDogs relocated. He scored 27 goals in his second season of 2008-09, second on the IceDogs, and earned an invite to the St. Louis Blues development camp.

Midway through 2009-10, Agozzino was named Niagara captain after Alex Pietrangelo was traded to Barrie and finished the season with team highs of 37 goals and 66 points. Despite the success, Agozzino went undrafted in the NHL and signed an amateur tryout with the Blues' American Hockey League affiliate in Peoria. He returned to Niagara in 2010-11 and led it in goals for two straight seasons, including 40 goals, 48 assists and 88 points to finish fifth in the OHL scoring race.

Agozzino played his first professional season for Lake Erie of the AHL in 2012-13 after signing a one-year contract. He scored his first goal and the game-winner on opening night, a 2-1 win against Oklahoma City on Oct. 12, 2012, and had 52 points (20 goals, 22 assists) to earn a two-year, entry-level contract with the Colorado Avalanche. His first NHL recall was Nov. 21, 2014 and he made his NHL debut with an assist for the Avalanche in a 4-3 win against the Carolina Hurricanes the next night. Agozzino became a restricted free agent after the season and filed for salary arbitration but settled before the hearing by agreeing to a one-year, two-way contract.

Agozzino left the Avalanche organization as a free agent when he agreed to a one-year, two-way contract with the Blues on July 1, 2016 and was assigned to start the 2016-17 season with Chicago of the AHL. He returned to the Avalanche the following season and played 11 NHL games in 2018-19 before signing a two-year, two-way contract with the Pittsburgh Penguins on July 1, 2019. After Agozzino had two assists in 17 games for Pittsburgh, he was placed on waivers on Feb. 23 and claimed the next day by the Anaheim Ducks.