After drawing the first leg 1-1 back in Turin, the reigning Italian champions knew they had to score against their Serie A rivals in Florence to stand any chance of progressing.
It was a tense encounter at the Stadio Artemio Franchi but, after Viola defender Gonzalo Rodriguez was sent off for picking up a second yellow card in the 71st minute, Pirlo was able to unlock the game and fire Juve through with a trademark free-kick.
It was the second time in five days that the veteran playmaker had earned his side a 1-0 away victory in such a fashion - he whipped home a free-kick at Genoa on Sunday.
"We knew that it was going to be a long battle and that we needed to score," Pirlo told juventus.com.
"Despite those first few minutes when we were under pressure, we had prepared well and we believed we could do it all the way.
"We were aware that Fiorentina would put on a good show, but since we dropped out of the Champions League our objective has always been to do well in the Europa League and Serie A.
"We're pleased with how it's going.
"It was important to win both games and I'm happy to have unlocked them with my goals."
Juve next face either Lyon, Basle, Sevilla, Benfica, AZ Alkmaar, Valencia or Porto and will find out who stands in their way when the last-eight draw is made on Friday.
Pirlo added: "I don't have any preferences on who we come up against. For now we'll enjoy this victory and we'll turn our attentions to the next game when the time comes."