Visitors Valencia needed at least two goals to progress and Jonas's powerful finish 10 minutes into the second half gave PSG something of a scare.
Valencia had looked very much out of this last-16 tie when they were 2-0 down going into the final minute of the first leg last month, but after pulling a late goal back there, Jonas' strike left them needing just one more to complete a remarkable turnaround.
Alas, it was not to be for the Spaniards, as Ezequiel Lavezzi - who had also scored at the Mestalla - levelled on the night in the 66th minute and PSG held out to secure their place in the last eight for the first time since 1995.
Given the ambitions of the French capital side's mega-rich Qatari owners, failing to make it through would have been nothing short of a disaster, especially given the way they played for most of the first leg.
"We deserved our qualification over the two legs," said PSG coach Carlo Ancelotti.
"We didn't play in a relaxed manner in the first half, but paradoxically it was after Valencia scored that we started to play and cause them problems."
Valencia coach Ernesto Valverde was visibly disappointed with the result but admitted that he could not criticize his players for their performance.
"It was a hard-fought game," he said. "We wanted to score the first goal and make them nervous, but in the end it wasn't to be.
"I am satisfied with the way the team played, even if I'm not happy to be out."
Just as in the first leg, Ancelotti's side seemed happy for Valencia to have most of the possession, but they lacked the same punch on the breakaway this time and almost paid dearly for inviting Valencia on to them.
Despite being deprived of Zlatan Ibrahimovic due to suspension following his sending-off at the Mestalla, Ancelotti opted not to hand a first Champions League start of the season to Kevin Gameiro, and PSG lacked any kind of reference point in attack as a result.
David Beckham, meanwhile, started on the bench and that was where he stayed for the night.
The first real chance of the game fell the Spanish side's way, with Roberto Soldado testing goalkeeper Salvatore Sirigu quarter of an hour in.
PSG were struggling to find any rhythm, and were disrupted even more when Christophe Jallet was forced off before the half-hour mark, seemingly due to an injury, with Gregory Van der Wiel taking his place.
But for all their dominance of possession, Valencia could not seriously trouble Sirigu, with Jonas and Tino Costa both missing the target before the Brazilian had a powerful shot beaten away by the keeper just before the interval.
For PSG, that served as a warning of what was to come, with Valverde signalling his intent by throwing on creator Ever Banega for midfield destroyer David Albelda at half-time.
He was rewarded for his positivity when Jonas struck soon after the restart, his powerful effort from 20 yards surprising Sirigu at his near post.
Suddenly a tie that had appeared to be limping towards a tame stalemate had come to life, and Ancelotti finally turned to Gameiro.
The striker, who Valencia had wanted to sign in 2011 when PSG lured him from Lorient, looked lively as soon as he was thrown on for midfielder Thiago Motta in the 58th minute, and he made PSG's equalizer on the night just before the midway point in the second half.
After driving through the middle of the Valencia defence, he got the break of the ball, but it was Lavezzi who shot, and then converted at the second attempt after 'keeper Vicente Guaita had blocked his initial effort.
It was Lavezzi's fifth goal in the Champions League this season, confirming that PSG do not rely solely on Ibrahimovic to score when it really matters.