The French club progress on the away-goals rule after the tie ended 3-3 on aggregate.
In an ill-tempered match, PSG had suffered an early setback when Zlatan Ibrahimovic was wrongly shown a straight red card for a challenge on Oscar.
Chelsea looked to have wrapped up the tie in the 81st minute when Gary Cahill blasted home after the visitors failed to clear a corner.
But PSG showed their resolve and leveled just five minute later when former Blues defender David Luiz powered a header past Thibualt Courtois to take the game into extra-time.
Chelsea again went in front – Eden Hazard scoring from the spot after Thaigo Silva’s handball – before Courtois pulled off an amazing save to keep out Silva’s header.
But from the resulting corner in the 114th minute the Brazilian outjumped both John Terry and Cahill to loop a header into the far corner.
PSG now join Real Madrid, FC Porto and Bayern Munich in the quarterfinals.