Zlatan Ibrahimovic was dismissed in the first half of the last-16 second-leg clash for a challenge on Oscar, then Gary Cahill fired the hosts in front late on before David Luiz haunted his former club with a leveller four minutes from time to force the extra half hour.
Early in the extra period, Eden Hazard slotted home from the spot after Silva was penalised for handball, but the PSG captain redeemed himself as he headed in a late goal to seal a 3-3 aggregate result and put his side through.
It was far from a convincing performance from the hosts, and Laurent Blanc’s men go into the last eight after getting revenge for their defeat to the Blues in last year’s competition.
In a first half lacking any real clear-cut opportunities, it was other matters which dominated the opening period. Diego Costa and Luiz resumed their spat from the first leg and spent plenty of time locked in a physical battle.
Half an hour in, Ibrahimovic went in for a loose ball against Oscar and the Brazilian came off worse. As the Chelsea players surrounded the referee in anger he showed Ibrahimovic a straight red card, leaving Blanc’s men with a long time to play a man down.
Referee Bjorn Kuipers’ work was far from done and he made another big call as half-time approached. Costa picked the ball up on the edge of the box and showed some neat footwork to squirm his way past two defenders before going down under a challenge from Edinson Cavani, but the Dutch official immediately waved Chelsea’s claims away.
Mourinho’s men struggled to find their rhythm after the break, and they were nearly punished as the hour approached, Cavani latching on to a ball through the middle before rounding Thibaut Courtois, but he could only find the inside of the near post as it flew wide of the target.
The fouls and scuffles continued throughout as Chelsea tried to stop PSG gathering any momentum, and nine minutes from time they found themselves in front.
Thiago Silva could only head a Blues corner up into the air and, after John Terry and Costa couldn’t get a clean effort away, it fell for Cahill, who smashed a volley into the roof of the net to make it 1-0.
But their lead would only last five minutes as Courtois was forced to concede a corner at the other end, and Luiz fired in a bullet header which flew past the goalkeeper and in off the woodwork to force extra time.
Five minutes in Chelsea were back in front after Luiz met Costa’s cross but could only head it up into the air, and Silva inexplicably went up for it with his arm, giving a simple decision for the referee to point to the spot.
Hazard’s coolness on the spot paid dividends for the hosts as it has done so often before, and he outwitted Salvatore Sirigu to send it low to the goalkeeper’s left and put Chelsea 2-1 up.
But PSG captain Silva would have the last laugh, seeing one header turned behind after a stunning save by Courtois before he looped another beyond the Chelsea shot-stopper seconds later, making it 3-3 on aggregate but giving the visitors the edge on away goals.
The visitors, who went out to Chelsea on away goals after a 3-3 aggregate draw last year, held on to seal a vital win for Blanc in dramatic fashion.