Ernesto Valverde's side made a valiant effort to overturn a 2-1 deficit from the first leg, taking a 55th-minute lead against the wealthy Qatar-backed French club through Jonas.
PSG struck back 11 minutes later when a mistake in midfield led to Ezequiel Lavezzi's leveller and Valencia were unable to breach the home side's defences again and exited 3-2 on aggregate.
"We are sad because we played well even though we needed to create more chances," Jonas told reporters.
"We hope the fans will get behind us between now and the end of the season so we can return to the Champions League," added the Brazilian forward.
Valencia are fifth in La Liga with 12 matches of the season remaining, a point behind Malaga who occupy Spain's fourth and final Champions League berth.
Getting back into the competition next term is particularly important for the club, one of many in La Liga struggling to stay afloat after years of financial struggles.
Rather than splurging hundreds of millions of euros on top players like PSG, Valencia have been forced to offload key performers like David Villa, David Silva and Juan Mata.
They are now effectively owned by the regional government after the club's foundation was unable to keep up with payments on a bank loan guaranteed by the Valencia administration that was used to buy a stake in the club.
In the 2010-11 season, Valencia had debts of almost 400 million euros, according to a recent study by an accounting professor at the University of Barcelona, while a new stadium under construction has been left half built.
Valencia visit Athletic Bilbao in La Liga on Sunday and will need to show more punch in attack against the Basques than they managed in Paris on Wednesday.
"We were lacking in clarity in the final part of the pitch, in electricity and speed," Valverde said.
"We are a bit down but we know that the team played well and we are leaving with heads held high despite the elimination."
- Sports & Recreation
- La Liga
- Champions League