Faced with the prospect of playing more than an hour with 10 men after Alex Telles was sent off for two yellow cards in as many minutes, Porto coped well as Juve saw a host of chances go begging.
The game appeared set to end a highly frustrating draw for Juve, but Massimiliano Allegri - who dropped centre-back Leonardo Bonucci in the wake of their spat at the end of Friday's win over Palermo - made a pair of inspired changes in bringing on Pjaca and Alves.
Pjaca finally broke the deadlock in 72nd minute, powering a half-volley into the bottom-left corner to end his wait for a maiden Bianconeri goal.
And just two minutes later Alves put Juve in complete command by converting from Alex Sandro's left-wing delivery to the far post.
The hosts never looked like mounting any sort of response and they now have a mountain to climb in the return fixture on March 14.
Porto started brightly and saw Yacine Brahimi go close when he curled a free-kick narrowly over after Giorgio Chiellini was dubiously adjudged to have fouled Andre Silva.
Juve soon began to dictate play without creating much in front of goal, Paulo Dybala spurning their best chance as he fired over on the stretch from Gonzalo Higuain's flick-on.
And the visitors were handed a huge boost as Telles lost all composure and committed two rash fouls in the space of as many minutes.
After raking his studs down the back of Juan Cuadrado's leg, the left-back needlessly slid into Stephan Lichtsteiner on the right flank, leaving referee Felix Brych with no choice but to issue a second yellow.
Cuadrado then immediately drove wide of the left-hand post before Miralem Pjanic forced Iker Casillas into action with a long-range effort.
Sami Khedira could only manage to turn Alex Sandro's left-wing cross wide, with Casillas then required to pull off a fine reaction save after Higuain's close-range effort deflected off Felipe.
The Juve onslaught continued and only the left-hand post prevented Dybala from breaking the deadlock with a fierce effort that had Casillas easily beaten.
Dybala did beat Casillas shortly after the restart when he lashed in at the near post from a clever Pjanic pass, only to be denied by the offside flag.
Juve were almost punished for their profligacy by Hector Herrera, who perhaps should have done better as he nodded wide from substitute Miguel Layun's excellent cross from the left.
A dipping volley from Khedira was only just off target and Higuain saw a sublime curling effort drift agonisingly wide as Juve's frustration grew.
But Allegri was able to breathe a sigh of relief when Pjaca lashed in after Layun inadvertently deflected Dybala's pass into his path.
Moments later Alves made an instant impact as he stretched to turn in the ball from the excellent Alex Sandro and, although Khedira missed a late chance to make it 3-0, Juve will head back to Turin extremely confident of progression to the quarter-finals.