Pittsburgh led by 20 points with 7:22 left before UNC's frantic comeback behind Marcus Paige, who scored 20 of his 27 points after halftime before fouling out with 25.4 seconds left.
UNC got it to 78-75 on Nate Britt's jumper with 11.4 seconds left but couldn't complete the comeback. James Robinson knocked down two free throws to make it a two-possession game again right after Britt's score, then Lamar Patterson rebounded a hurried 3-pointer by James Michael McAdoo with 4.3 seconds left to essentially seal it.
Now, after a dominating performance in Thursday's second round and a bumbling-but-successful finish Friday, the Panthers will face sixth-ranked and top-seeded Virginia in Saturday's semifinals.
Zanna, a 6-foot-9 fifth-year senior, came within two rebounds of tying the single-game tournament rebounding record that had stood since 1959. He fouled out with 1:03 left.
Zanna had 10 offensive rebounds - one shy of UNC's team total - and helped Pitt control the boards while taking a 21-6 edge in second-chance points against a team that was third in the league in rebounding margin for the regular season.
With all those point-blank extra looks off offensive rebounds, Pitt shot 52 percent for the game.
North Carolina spent all day trying to recover from a terrible start. It missed 16 of 19 shots and fell behind 25-8, though its late comeback at least gave them some hope of sticking around Greensboro a little longer.
North Carolina shot 42 percent, though much of that production came after Pitt had built a 61-41 lead on Josh Newkirk's three-point play midway through the second half.
The Panthers have won five of six to pull out of a three-game losing streak that started to threaten their NCAA tournament chances, with this ranking as probably their best of the year.
Now the Tar Heels are suddenly struggling.
The Tar Heels started 0-3 in the ACC and stood at 1-4 before running off 12 straight wins, including a home win against Pittsburgh, to earn a top-four seed and double-round bye that came with it. But UNC didn't play well in the final two games of the streak and lost at Duke to close the regular season last weekend, sending them into Greensboro coming off their first loss since Jan. 20.
The rebounding trouble was particularly alarming for UNC considering it managed just 20 rebounds in last weekend's loss at Cameron Indoor Stadium.
During one demoralizing second-half stretch, Zanna tipped in offensive rebounds on consecutive possessions and had 17 rebounds to UNC's 16 with nearly 14 minutes still left.