Following a 1-1 draw in the first leg, Pirlo scored the only goal at the Stadio Artemio Franchi in the 71st minute by firing home from just outside the area.
That came moments after Fiorentina captain Gonzalo Rodriguez had collected a second yellow card for hauling down Fernando Llorente and it proved to be the defining moment of the contest as La Viola could not find a way back.
They came into the contest with a distinct advantage as Mario Gomez's late strike gave them an important away goal in the first leg.
And while Juve were 27 points clear of their opponents in Serie A, it was the home side that came charging out of the blocks, with Gomez seeing his left-foot strike from the edge of the area trickle past the post.
The Italian league leaders were then given another let-off when Juan Guillermo Cuadrado was given plenty of space on the right but no Fiorentina attackers were on hand to provide the final touch to his dangerous low cross.
That seemed to galvanise Juventus, who started to find some rhythm as the half wore on and their pressing of Fiorentina off-the-ball nearly paid dividends when goalkeeper Murara Neto only just cleared the danger from Cuadrado's sloppy backpass.
Paul Pogba did well to chest down from a ball over the top, but Rodriguez did just enough to put the French midfielder off his stride and his shot with the outside of his right foot went harmlessly wide.
Yet La Viola ended the half strongly, although Josip Ilicic was wasteful when he shot over after Giorgio Chiellini had inadvertently headed into the Slovenian's path.
Fiorentina - who handed Juve their first, and so far only, league defeat of the season when they triumphed 4-2 in Florence last October - started well after the interval.
Borja Valero's free-kick from just outside the area worked Juventus goalkeeper Gianluigi Buffon, who pushed away Cuadrado's swerving shot.
At the other end Arturo Vidal, who scored Juve's goal in the first leg, missed a gilt-edged chance after putting a free header over from Mauricio Isla's right-wing cross.
But the game changed in an instant in the 69th minute when Fiorentina were reduced to 10 men after skipper Rodriguez received his marching orders for a rash challenge on Llorente on the edge of the area.
And from the resulting set-piece, free-kick specialist Pirlo expertly thumped the ball into the top right-hand corner, past the outstretched hand of Neto.
The Fiorentina goalkeeper made amends with a fantastic point-blank save from Llorente's header and again at his near post from Stephan Lichtsteiner's strike.
Vidal should have wrapped it up as Juve dominated in the closing stages and although he headed wide of the target with the goal gaping, it mattered little to the overall result.