Arturo Vidal and Leonardo Bonucci had put the champions and league leaders 2-0 ahead before second-placed Roma received two dismissals in quick succession with around 15 minutes remaining.
First, Daniele De Rossi saw red for a high challenge on Giorgio Chiellini before Leandro Castan was dismissed for handball on the goalline.
Former Roma man Mirko Vucinic applied the third from the penalty spot as Juve started 2014 in the same ominous fashion they ended last year, ending Roma's 19-match unbeaten league record in the process.
In an opening period devoid of clear-cut chances, Juve found the breakthrough when Vidal collected Carlos Tevez's pass and applied the finish from an acute angle 17 minutes in.
Antonio Conte's men - who have not suffered defeat in Serie A since a 4-2 loss to Fiorentina in October - essentially put the game out of reach three minutes after the break when Bonucci slid in to convert Pirlo's whipped free-kick.
Roma then completely unravelled as De Rossi and Castan were sent off, before Vucinic rounded off the scoring from 12 yards.
The win sees Juventus open up an eight-point gap on Roma at the top of Serie A, strengthening their hopes of claiming a third consecutive title.
Pirlo returned to Conte's team for the first time since the start of December after recovering from a knee injury, while Roma named a familiar front three with Adem Ljajic and Gervinho assisting Francesco Totti.
Juve initially struggled to impose themselves in the game, but went ahead with their first effort on goal after 17 minutes.
Tevez received a pass inside the area and flicked a smart throughball into the path of Vidal, who squeezed home past Morgan De Sanctis at the near post.
Roma were somewhat unfortunate to find themselves behind and pushed for a leveller, with Gianluigi Buffon on high alert to fend off Dodo's low long-distance strike.
The champions finished the half brightly, though, with Fernando Llorente coming close with a fierce 25-yard strike that went narrowly over the crossbar.
Juve's momentum continued into the second period, and it took just three minutes for their advantage to be doubled as Bonucci side-footed home Pirlo's teasing back-post delivery.
The home side were then dealt a blow when Tevez had to be replaced by Vucinic on the hour mark after appearing to tweak his hamstring.
But Roma's loss of discipline helped them, as De Rossi was sent off for scything down Italy team-mate Chiellini, before Castan was shown a straight red for handballing on the goalline from the resulting free-kick.
And Vucinic duly wrapped the match up on 77 minutes by finding the bottom corner from the spot.
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