Corelli had succeeded in gaining international reputation within no less than two years of his professional debut. In May 1953 he was chosen for the first Italian production of Prokofiev's War and Peace (Pace e Guerra) at the Maggio Musicale Fiorentino, and during the summer of 1953 he starred in Pagliacci as Canio in Rome, as Don Jose in Carmen in Bolzano, Enna and Ravenna, as well as a single performance of Pagliacci in Spoleto. In November he reappeared with Maria Callas as Pollione in Norma, followed by further performances of Norma throughout 1954 in Modena and Parma.

In March 1954 he debuted in Don Carlo to excellent reviews in Rome and was selected for the premiere of Spontini's Agnese Di Hohenstaufen in May in Firenze. In September he recorded Pagliacci for RAI in Milano before making his debut at La Scala 7 December 1954, in Spontini's La Vestale opposite Maria Callas. The performance received rave reviews and Corelli came to be dubbed the "Sputnik Tenor" for his swift rise to popularity throughout Italy.

In 1955 he participated in a black and white film of Tosca, edited from RAI in Milano and broadcasted on television. A year later he starred in a full colour version, from Roma, starring Franca Duval as Tosca, Corelli as Cavaradossi and Afro Poli as Scarpia, with the voices of Maria Caniglia (Tosca), Corelli (Cavaradossi) and Gian Giacomo Guelfi (Scarpia). There was further one television broadcast from Milano, this time of Turandot, produced by RAI and televised 23 December 1958. The cast included Corelli as Calaf, Lucilla Udovich as Turandot and Renata Mattioli as Liù.

Corelli was triumphant in Italy and became the leading spinto tenor at La Scala and Roma during the 1950's, where he appeared most frequently and to utter success in roles like Don José (Carmen), Mario Cavaradossi (Tosca), Calaf (Turandot), Manrico (Il Trovatore), along with the surprising and much talked about performances of Händel's Giulio Cesare in Egitto in Roma December 1955 and January 1956 and later at La Scala in December 1956; Händel's Hercules at La Scala in December 1958 with Schwarzkopf, Barbieri, Bastianini and Hines; Bellini's Il Pirata opposite Maria Callas at La Scala, May 1958; Adriana Lecouvreur in November 1959 with Olivero, Bastianini and Simionato, in Napoli and the triumphant Poliuto at La Scala in 1959 with Maria Callas.

His UK (London) debut came in 1957 with Tosca at Covent Garden, starring soprano Zinka Milanov as Tosca. It was Corelli's most important appearance outside Italy to date and prepared the ground for Corelli's forthcoming success in America. The Tosca at Covent Garden followed the described motion picture of 1956, which partly introduced the tenor to the American public. However, it was not until January 1961 that he actually performed in the US, when making a joint debut at the Met with soprano Leontyne Price. The occasion was Il Trovatore for the commemoration of the 60th anniversary of the death of Giuseppe Verdi. The reception was overwhelming with standing ovations carried on for almost an hour after the performance, and Corelli fully conquered the Met public with celebrated performances that very season in Andrea Chenier and Turandot, the latter revived after nearly 30 years of absence at the house.

After splendid performances in La Gioconda in New York 1962, he left for Salzburg and the Summer Festival for a staging of Il Trovatore. The invitation had come from Herbert von Karajan. Salzburg was left breathless. He further stunned opera lovers with a triumphant Gli Ugonotti (Meyerbeer) at La Scala, perhaps his most difficult role to date. It was a role thought entirely unsuited to him, given the dark hue of his voice and the horridly high tessitura of the part. Andrea Chenier opened the Met season in October and the Corelli decade at the Met had begun.