Studied, St. Pius College, Bombay. Bachelor in mathematics & pedagogy, State UNiv., Bombay. D in canon law, Pontifical Urban University, Rome, 1954.
Pimenta was ordained on 21 December 1949. Before coming to Rome to study at the Pontifical College of Saint Peter the Apostle, he served as assistant parish priest in Bombay and worked with the Bombay Curia. On 5 June 1971 Pope Paul VI appointed Pimenta as an Auxiliary Bishop of Bombay, giving him the titular see of Bocconia.
He was chosen to become the next Archbishop of Bombay by Pope Paul and was thus appointed Coadjutor Archbishop of Bombay on 26 February 1977.
On 11 September 1978 he succeeded Valerian Gracias as the Archbishop of Bombay. He has served as President of the Indian Episcopal Conference.
He was created Cardinal-Priest on 28 June 1988 by Pope John Paul II with the titular church of Santa Maria Regina Mundi a Torre Spaccata. He became Archbishop Emeritus of Bombay on 8 November 1996 with the appointment of Ivan Dias as his successor in that see.
He died on 19 July 2013.