He was ordained to the priesthood by Archbishop Giuseppe Ceppetelli on 14 June 1912, and then did pastoral work in Rome until 1920.

Grano was named a pontifical ceremonery supernumerary on 23 January 1920, and also served as a staff member (1923–1945), chief of protocol (1945–1953), and Substitute (1953–1958) of the Vatican Secretariat of State.

On 13 December 1958, Grano was appointed Apostolic Nuncio to Italy and Titular Archbishop of Thessalonica.

He received his episcopal consecration on the following 27 December from Pope John XXIII himself, with Bishops Girolamo Bortignon, OFM Cap, and Gioacchino Muccin serving as co-consecrators, in St. Peter's Basilica.

Pope Paul VI created him Cardinal-Priest of San Marcello al Corso in the consistory of 26 June 1967.