The Qatari-backed club were far from convincing in the last 16 second leg and after a first half short on entertainment, Valencia made the tie interesting when Brazilian Jonas put the Spaniards ahead with a superb 25-metre strike on 55 minutes.
PSG, without suspended top scorer Zlatan Ibrahimovic who was sent off late in the first leg, equalised 11 minutes later when Lavezzi scored at the second attempt after his initial shot was parried back to him by keeper Vicente Guaita.
Another Valencia goal would have sent the game to extra time but PSG's defence, marshalled superbly by Thiago Silva - back from a two-month injury layoff - held firm.
The Ligue 1 leaders, who have not lost at their Parc des Princes home in 23 European matches since 2006, last played in the last eight of Europe's premier club competition in 1995 when they reached the semi-finals.
"The qualification is deserved, even if we struggled more than during the first leg," PSG coach Carlo Ancelotti told a news conference.
"I was a bit worried after their goal because it was wide open again. In the first half we controlled but played too deep.
"Their goal woke us up."
Asked whether PSG could win the competition, Ancelotti said: "We certainly are doing our best but I cannot say."
Only Javier Pastore's delicate turns caused trouble to the Valencia midfield as PSG's cautious approach gave the Spaniards the confidence to attack.
Roberto Soldado's weak attempt from a clever Jonas through ball was easily blocked by PSG keeper Salvatore Sirigu as Valencia carved out the first chance on 15 minutes.
Jonas had a fierce shot parried away on the stroke of half-time and Clement Chantome's corner kick that went straight out of play epitomised a woeful first half.
