The Brazilian centre-back gave away a penalty with a bizarre handball six minutes into extra time, but he made amends by heading a dramatic equaliser with six minutes left to play to send PSG through on away goals, with the tie ending 3-3 on aggregate.
The visitors had been facing an uphill task when star striker Zlatan Ibrahimovic was controversially sent off shortly after the half-hour mark, but they got off lightly when a foul on Diego Costa in the box went unpunished.
Edinson Cavani then hit the post for PSG before Gary Cahill looked to have settled an ill-tempered tie with an 81st-minute drive, before former Chelsea man David Luiz – who had earlier elbowed Costa out of sight of the referee and feigned being headbutted in a bid to get the Spain international sent off – forced an additional 30 minutes with a bullet header five minutes later.
PSG had the momentum heading into extra time, but they were undone when Thiago Silva attempted to palm the ball clear in the box having mistimed his jump, giving Eden Hazard the chance to fire the hosts in front from the spot.
The Belgian made no mistake as he coolly converted, but Thiago Silva had the last word – looping a header over Chelsea goalkeeper Thibaut Courtois to stun Stamford Bridge into silence.
Jose Mourinho made one change to the side that started the first leg, with Oscar coming in for Willian, while Thiago Motta slotted into the PSG midfield having missed the home tie through injury.
Ibrahimovic saw red for recklessly lunging into a challenge on Oscar in the 31st minute, although the Sweden international did look to be trying to withdraw from the tackle when contact was made.
The forward's case was not helped by the sea of blue shirts that surrounded referee Bjorn Kuipers - who officiated last year’s final - in the immediate aftermath of that flashpoint.
Oscar could count himself lucky to remain on the pitch when he avoided a yellow card for bundling into Marco Verratti a few moments after receiving his first booking, before Costa justifiably accused Luiz of catching him with an elbow.
Costa and Luiz squabbled like schoolboys for much of the encounter, and the former should arguably have been awarded a penalty two minutes from the break when his heel was clipped by Cavani in the box.
Cavani came agonisingly close to breaking the deadlock when he beat the offside trap to take the ball beyond Courtois in the 57th minute, but the Uruguayan's eventual effort from a tight angle struck the near post.
Courtois got down well to keep out Javier Pastore with 20 minutes left on the clock, before the match threatened to descend into farce.
First, Costa received a booking for a cynical challenge on Thiago Silva, before being confronted by chief nuisance Luiz.
The Brazil international recoiled as if he had been headbutted by the striker, but replays exposed his play-acting, spotted by Kuipers, who added Luiz’s name to his already-bulging notebook.
Sadly, the incident was indicative of a match that was marred by poor discipline, but Cahill looked to have put the tie to bed nine minutes from time.
The ball fell to the England defender as PSG failed to clear their lines from a corner, and he rifled home from 12 yards.
That lead lasted only five minutes, though, as Luiz found the net with a firm header from a corner, a few moments after claiming to have had his shirt pulled in the box.
Thiago Silva’s bizarre slapped clearance offered Chelsea a chance to apparently put the match to bed six minutes into the additional period – an opportunity Hazard was only too grateful to take from the penalty spot.
At that stage, Chelsea looked to be heading into the last eight, but having cost his side earlier in extra time, Thiago Silva made amends.
His looping header from a corner turned the match on its head, seeing PSG into the quarter-finals at the end of a frantic game and reversing the outcome of last year's last-eight meeting between the pair, when Chelsea had progressed on away goals after a 3-3 draw over two legs.
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