Born at Ezeiza, Tarantini rose through the Boca Juniors youth divisions in the early 1970s, and was noted for his afro hairdo and his large front teeth, which earned him the nickname conejo ("rabbit").

In 1977 with Boca Juniors he won his first international club football competition and one of the most prestigious tournaments in the world and the most prestigious club competition in Latin American football – Copa Libertadores, when after the goalless draw Boca defeated Cruzeiro 5–4 on penalties.

[1] Tarantini was part of the Argentina under-23 team that won the 1975 Toulon Tournament, together with Jorge Valdano, Américo Gallego, and others, with César Menotti as coach.

After his performances during the World Cup (he scored a goal in the 6–0 victory against Peru, and played in the final against The Netherlands) he was signed by Birmingham City for £295,000.

After his return to Argentina he played for Talleres de Córdoba, River Plate, and European teams SC Bastia, Toulouse and FC St. Gallen.