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."
Valencia were forced to make do with a makeshift defence with centre-backs Adil Rami and Ricardo Costa both injured, as well as full-back Joao Pereira who was ruled out after picking up a hamstring injury during training on Tuesday.
The visiting back four, however, were barely threatened by a disappointing PSG side in the first half on a mild Parisian night.
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.
Ancelotti was then forced to make a change when Gregory van der Wiel came on for injured right-back Christophe Jallet in the 27th minute.
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.
PSG were eventually punished when Jonas unleashed a powerful shot that went into the far corner, his fifth goal in the competition this season helping the game spring into life.
Ancelotti added weight to his front line by replacing midfielder Thiago Motta with striker Kevin Gameiro who had an almost immediate impact.
He dribbled his way into the area and as he was closed down by three defenders, the ball fell to Lavezzi who scored at the second attempt.
NO BECKHAM
Former England captain David Beckham, who signed a short-term deal with PSG in January, will have to wait to make his first Champions League appearance since 2010 with AC Milan after
