Eden Hazard's penalty had given the Blues a potentially vital away goal, but Javier Pastore's brilliant stoppage-time solo goal, added to earlier strikes from Ezequiel Lavezzi and a David Luiz own goal, saw PSG take a two-goal advantage into next week's second leg at Stamford Bridge.
Chelsea, the 2012 European champions, were looking to book their place in a seventh semi-final in 11 campaigns but started without a recognised striker.
With Samuel Eto'o injured, Jose Mourinho selected Andre Schurrle as a 'false nine', appearing to finally lose patience with Fernando Torres.
PSG, unbeaten in their previous 28 European home games, began with a bang and took a fourth-minute lead when one of the three strikers Mourinho had highlighted could be the difference in the contest struck a fine half-volley.
It was Lavezzi - rather than Zlatan Ibrahimovic or Edinson Cavani - who fired into the roof of the net after a poor headed clearance from John Terry.
Chelsea equalised before the interval when Hazard - courted by PSG - converted a penalty after Oscar was fouled by Thiago Silva.
Hazard also hit the post before half-time and PSG were denied a spot-kick when Cavani turmbled under Gary Cahill's aerial challenge.
Torres replaced Schurrle after 60 minutes, but no sooner had he joined the action than Chelsea were behind again as Lavezzi's free-kick was swung in menacingly and bundled into the net by David Luiz.
Ibrahimovic, still chasing a first Champions League triumph, departed clutching his right hamstring with more than 20 minutes remaining - but Pastore ensured the night and potentially the tie would end in PSG's favour, beating Petr Cech at his near post after a mazy run.
Two changes were made to the Chelsea line-up following the loss at Crystal Palace which saw Mourinho concede the Premier League title chase was over.
Ramires and Willian replaced the ineligible Nemanja Matic and Torres, whose place at the focal point of the attack was taken by Schurrle.
The Germany international performed the same role against Manchester United last August in a dour goalless draw and Mourinho would have been content for a similar result.
That possibility was soon extinguished as Ibrahimovic robbed Luiz and Terry's weak header from Blaise Matuidi's cross fell for Lavezzi, who hit an exquisite half-volley into the roof of Cech's net.
Chelsea were struggling to get hold of the ball, giving it away freely, and a poor Schurrle corner resulted in a Lavezzi-led counter-attack and Ramires halting the move cynically at halfway.
The Brazil midfielder, in the middle of a domestic suspension, will miss next Tuesday's second leg as a result.
PSG were content for Chelsea to pass the ball at the back and - bar one Luiz shot from distance which was easily held by Salvatore Sirigu - there was little danger from the visitors.
The threat from PSG was all the more apparent, as Lavezzi held off Cahill to squeeze a shot into the side-netting.
Chelsea were awarded a penalty when Thiago Silva mis-timed his lunge towards Oscar, felling his compatriot.
Hazard sent Sirigu the wrong way from the spot to equalise.
The Belgium forward then beat Sirigu again only to strike the foot of the post with a fine volley.
PSG were denied after Ibrahimovic beat Terry to a cross from the right and nodded centrally towards Cavani, who hit the deck as he battled with Cahill.
On this occasion Serbian referee Milorad Mazic was unmoved.
PSG had slackened off after the opening goal and resumed after the break with renewed vigour, Lavezzi rising unopposed to head just over from Matuidi's left-wing cross.
With PSG pushing for a second, Mourinho realised Chelsea would have space on the break and replaced Schurrle with Torres with 30 minutes remaining.
Moments after Torres entered the fray the hosts had their second when Lavezzi's wickedly inswinging free-kick bamboozled everybody, particularly Luiz, who inadvertently kicked the ball into his own net.
Hazard made a fine burst as he tried to spark Chelsea, but Thiago Silva cut out his pass intended for Torres.
Willian then beat Maxwell on the Chelsea right, but no-one was in the box to react to his cross.
Ibrahimovic, after a subdued performance, limped off, with Cavani moving into a central role.
The Uruguay striker wrong-footed the entire Chelsea defence with six minutes to go and his fierce shot beat Cech but went past the post.
PSG claimed their third when Pastore danced around three Chelsea defenders on the right of the box, cut in and beat Cech with a fine shot.