After a poor start to the season, last year's double winners Juve have enjoyed a Serie A revival to move within six points of leaders Inter Milan with one game remaining before the festive break.
And Massimiliano Allegri's men provided some early Christmas cheer with a comprehensive victory over their bitter city rivals to maintain their chances of a second successive double.
"It was important for us to win this and get to the quarters, but now we're thinking of Sunday's game away to Carpi," Allegri told Rai.
Mandzukic, Dybala strike late as Juve stun Viola
"For the quality they have, they are an underestimated side."
Zaza, making a rare appearance amid reports of a January move to the English Premier League, put the hosts ahead with a cracking first-time strike on the half hour after great work on the left wing by Alvaro Morata.
The 24-year-old doubled Juve's lead six minutes after the restart when he guided a low shot beyond Salvador Ichazo and into the far corner.
Dybala has scored in Juve's past two league wins and made if three matches in a row with a well-taken strike in the 73rd minute after replacing Zaza eight minutes into the second half.
Dybala hits stunner as Juve sink sorry Lazio
Pogba wrapped up the win with a superb free-kick eight minutes from time to set up a quarter-final clash with the winner of the Lazio-Udinese tie on Thursday.
Both Dybala and Pogba celebrated their goals by lifting their shirts to show respective messages, the Frenchman's for 'Bryan'.
"He's an 11-year-old kid who has cancer, we promised we'd dedicate a goal to him," Pogba told Rai.
Pogba said his free-kick skills have been honed with the help of ex-Juve star Andrea Pirlo, now at New York City FC in the MLS.
Pogba, Evra and Khedira mentally fit to play - Allegri
"I'm working a lot on my free-kicks given that Pirlo's not here, in fact I have him to thank for teaching me so much."
Earlier, Napoli secured an exciting last eight tie with Inter after a 3-0 win over Serie A basement side Verona at the San Paolo, where Omar El Kaddouri, Dries Mertens and Jose Callejon got on the scoresheet.
The quarter-finals will be without two leading Serie A sides, however, as Roma and Fiorentina fell to unheralded opposition.
Serie A title hopefuls Roma fell 4-2 on penalties to Spezia after failing to find a way past the second division side after two periods of extra time at the Stadio Olimpico.
Under-fire coach Rudi Garcia hit out: "Today was a complete failure but now we need to keep our heads during the storm.
"Above all, we need to show guts, pride and much more besides to show we're capable of much better, starting from Sunday because this is not acceptable."
Fiorentina, who sit four points behind Serie A leaders Inter, suffered a 1-0 upset to second-from-bottom Carpi, for whom Antonio Di Gaudio scored a 76th minute winner.
Spezia will now meet third division Alessandria, who beat top-flight Palermo a fortnight ago and stopped Genoa's progress on Tuesday with a 2-1 win.
Carpi will now meet the winner of Sampdoria and AC Milan, who play in Genoa on Thursday.
Inter Milan progressed to the quarters on Tuesday with a 3-0 home win over Cagliari.
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