He was born in Ancona, the son of a ship worker. He studied at the Pesaro Conservatory of Music. Largely self-taught, his singing style was modelled on recordings by Caruso and Beniamino Gigli and struck professional critics as self-indulgent and unnecessarily athletic. His innate musical intelligence brought refinement as his career developed. His French was never idiomatic, but he infused even the warhorses of Neapolitan songs like 'O Sole Mio' with freshness and authenticity.
In 1951, he won the Maggio Musicale in Florence, earning a debut at the Spoleto Music Festival, where he sang Don José in Carmen. He debuted at the Rome Opera in 1953 in Riccardo Zandonai's Giulietta e Romeo, and quickly became a steadfast member of the company, with an active repertory of some thirty roles.
In 1958 he married Loretta di Lelio, a fan who became his public relations agent.
Corelli made his debut at New York's Metropolitan Opera on 27 January 1961 as Manrico in Il Trovatore, in a double debut with Leontyne Price. Critic Harold Schoenberg noted Corelli's "exciting animal drive" and his need for some refining polish. Later that season Corelli and Birgit Nilsson put Turandot back in the standard repertory there. He eventually sang nineteen roles in fifteen seasons and became a fixture at the Met.
Despite his virile, "heroic" stage presence, Corelli suffered from terrible stage-fright. "They had to push him on stage," the soprano Renata Scotto recalled.
Corelli retired from the stage in 1976 at the young age of 55,
He died in Milan in 2003, having suffered a stroke earlier that year.