Barinaga joined Patronato from Colón in 2016. After four years progressing through their youth ranks, Barinaga made the breakthrough into Gustavo Álvarez's first-team squad in 2020. He initially featured during pre-season, notably scoring in a friendly match with Atlético Paraná on 11 January. Barinaga made his senior debut on 14 December 2020 during a 4–0 defeat in the Copa de la Liga Profesional away to Rosario Central, as the midfielder replaced Fernando Luna with thirteen minutes left.
