The French champions waited till the 93rd minute for substitute Eric Choupo-Moting to score after Marquinhos equalised in the 90th minute.
Mario Pasalic gave the Italians a 27-minute lead, which was the only goal until the 90th minute.
The match was going to extra time before Choupo-Moting scored to send the French champions to the semi-finals in the Champion League, for the first time since 1995.
They will play RB Leipzig or Atletico Madrid next Tuesday for a place in the final.
It is the first time PSG have gone beyond the last eight of the Champions League since the Qatari takeover of the club almost a decade ago and all the huge investment that has followed.