Rodrigues played four seasons at Boston University, where he scored 121 points (42 goals, 79 assists) in 146 games. As a senior in 2014-15, he scored 61 points (21 goals, 40 assists) in 41 games to finish second in NCAA Division I scoring to linemate Jack Eichel.

An undrafted free agent, Rodrigues signed a two-year, entry-level contract with the Buffalo Sabres on April 22, 2015, two months before the Sabres selected Eichel with the No. 2 pick of the 2015 NHL Draft.

Rodrigues played most of the 2015-16 season with Rochester of the American Hockey League but was called up late in the season. He made his NHL debut April 8, 2016, against the Columbus Blue Jackets, and scored his first NHL goal one night later, against the New York Islanders.

He signed a two-year contract with the Sabres on July 27, 2017, but a broken hand limited him to 48 games. By December he had established himself as a regular among Buffalo's bottom-six forwards, with the potential to move up in the lineup, and he finished with 25 points (seven goals, 18 assists).

Rodrigues had 29 points (nine goals, 20 assists) in 74 games in 2018-19. He was traded to the Pittsburgh Penguins on Feb. 24, 2020.

Rodrigues signed a one-year contract with the Colorado Avalanche on Sept. 12, 2022. He left as a free agent to sign a four-year contract with the Florida Panthers on July 2, 2023, and played a key role in helping the Panthers win the Stanley Cup for the first time in 2024 and again in 2025.