Noted for his technical ability as well as stamina, Tarantini was best known for his fiery temper.
He 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").
Tarantini was part of the under-23 team that won the 1975 Toulon Tournament, together with Jorge Valdano, Américo Gallego, and others, with César Menotti as coach. He became the left-back of the Argentina national football team after Jorge Carrascosa left the team (the book El DT del Proceso by Gasparini and Ponsico claims that the wolf Carrascosa declined to play for the dictatorship). He was also, at 22, the youngest player of that team.
A few months before the 1978 FIFA World Cup, he had a contractual dispute with Boca that left him clubless, as Boca management pressured all Argentine clubs into denying him a new contract. After some good performances during the World Cup (he scored a goal in the 6-0 victory against Peru) he was signed by Birmingham City for £295,000. His spell in England was overshadowed by poor discipline, with Tarantini flattening Manchester United striker Brian Greenhoff, and famously ending his 23-game spell in Birmingham by wading into the crowd to punch a heckler.
After his return to Argentina he played for Talleres de Córdoba, River Plate (where he had good performances as a full-back), and European teams SC Bastia, Toulouse and FC St. Gallen.
In 1982 Tarantini was voted into the Top Ten of the South America Player of the Year awards.
Tarantini also played in the 1982 FIFA World Cup for Argentina, and retired from the national team immediately there after.
Tarantini's love of nightlife was noted by gossip columnists; for many years he was the husband of fashion model Patricia Pata Villanueva.