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").
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 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. His spell in England was overshadowed by poor discipline, with Tarantini flattening Manchester United defender Brian Greenhoff, and ending his 23-game spell in Birmingham by wading into the crowd to punch a heckler.
^"Boca Juniors 1–0 River Plate". Rec.Sport.Soccer Statistics Foundation. Prepared and maintained by Pablo Ciullini and Osvaldo José Gorgazzi for the Rec.Sport.Soccer Statistics Foundation. 2 September 2009. Retrieved 15 February 2017. Campeonato Nacional 1976 (National Championship)
^"La Selección de Todos los Tiempos" [The Team of All Time] (in Spanish). Argentine Football Association. 4 January 2016. Archived from the original on 14 August 2018. Retrieved 29 January 2018.