Pirlo's dead-ball accuracy on 71 minutes, caressing the ball high into the far right of the net, handed Juve a 1-0 victory in Florence and a 2-1 aggregate success.
Valencia, Olympique Lyon and AZ Alkmaar also progressed to the last eight from the early kickoffs.
Lyon eased through 5-3 against Viktoria Plzen despite a 2-1 second-leg defeat in the Czech Republic while a second-half goal from Paco Alcacer earned Valencia a 1-0 win at home to Bulgaria's Ludogorets for a 4-0 aggregate win.
Dick Advocaat guided Zenit St Petersburg to the 2008 UEFA Cup title and his AZ Alkmaar side progressed to the last eight following a goalless draw in Ramenskoye with Anzhi Makhachkala.
Anzhi, bottom of Russian Premier League with a solitary win in 21 games, lacked the firepower to overturn a 1-0 first-leg defeat and their frustrations boiled over in stoppage time when Karlen Mkrtchyan was shown a straight red for a poor challenge on Steven Berghuis.
Ten-man Basel ended the impressive campaign of Austria's Salzburg but their 2-1 away win, after a goalless first leg, was marred by a first-half interruption of about 12 minutes after fans of the Swiss side threw objects on to the pitch.
Sevilla beat Liga rivals Real Betis 4-3 on penalties to complete a remarkable comeback.
Los Rojiblancos reversed a 2-0 first leg deficit to force extra time, before clinching victory in a tense shootout.
- Sports & Recreation
- Fernando Llorente
- AZ Alkmaar
- Viktoria Plzen
- Gonzalo Rodriguez
- Olympique Lyon