Llorente was handed his first Champions League start by manager Santiago Solari, filling in for the injured Brazil international Casemiro.
And the 23-year-old gave a mature performance at the base of the Madrid midfield as second-half goals from Gareth Bale and Vazquez earned victory at the Stadio Olimpico on Tuesday.
Llorente completed 91.1 per cent of his passes, a higher accuracy rate than team-mate Luka Modric, and Vazquez was full of praise for the academy product.
“Llorente has played a spectacular match,” Vazquez said after scoring his first goal of the season.
“It was not easy for this match, he had a lot of personality and a lot of enthusiasm. We have to be very happy with him.
“The matches are getting more difficult, this was very close and could have been won by anyone. The [first] goal gave us confidence.”