But John Mikel Obi scored a precious away goal, sweeping home a deserved leveller from six yards as he capitalised on some terrible defending.
Chelsea's stoic defending seemed on course to secure a draw before Cavani steered home Angel di Maria's pass.
The 29-year-old Uruguay striker, who has been used as back-up to Ibrahimovic this season, scored moments after substitute Oscar spurned a chance for the visitors.
But with the second leg to come at Stamford Bridge on 9 March, the Blues are still very much in contention thanks to Mikel's goal - and Thibaut Courtois' late save from Ibrahimovic.
Tie perfectly poisedSome have called this a make-or-break week for Chelsea - this match followed by an FA Cup tie with Manchester City on Sunday.
With their Premier League title defence already over, Guus Hiddink's men will know by the end of the weekend whether they can salvage anything from a disappointing domestic season.
But whatever happens against City, they will be hopeful of progressing in this competition after a promising performance in the French capital.
Blues step up to the challengeThis was the first defeat since Hiddink took over after Jose Mourinho's sacking in November, leading his team to a 12-match unbeaten run.
Chelsea thumped Newcastle 5-1 at the weekend, but Laurent Blanc's men were a step up in class.
The Ligue 1 leaders are unbeaten in 44 league games and lead second-placed Monaco by 24 points, though before the match Hiddink questioned the quality of the French top flight.
There is no doubting the quality of players at PSG's disposal, however, with Blanc able to recall the likes of of Ibrahimovic, Marco Verratti and Lucas Moura after resting them at the weekend.
The hosts were by far the superior side in the opening 17 minutes, having 74% of possession, but all they had to show for it was a long-range Lucas strike.
As the visitors dragged themselves back into the match, Ibrahimovic scored against the run of play. Indeed, were it not for Kevin Trapp's fingertips diverting Diego Costa's header onto the crossbar, the Blues would have been ahead before the Swede scored his first goal against Chelsea.
Ibrahimovic's free-kick struck Mikel en route to beating Courtois, but the Nigeria midfielder quickly made amends by capitalising on poor PSG defending from a corner.
Chelsea manage without Terry
With John Terry and Kurt Zouma injured, Chelsea were without their first-choice centre-backs. But the makeshift partnership of Gary Cahill and Branislav Ivanovic withstood wave after wave of second-half PSG attacks until eventually caving in in the closing stages.
Cahill bravely blocked a Blaise Matuidi shot that would have hit the net had the Englishman not put his body in the way.
The defensive duo were ably supported by Courtois, who made a series of saves, though he was culpable for allowing Cavani to thread the ball between his legs and in at the near post.