Cavani's goal came after 14 minutes following a bad defensive error but Lasse Schone curled home a free kick in the 74th minute to earn Ajax a point.
"We are certainly pleased with the draw. If you look at the mistakes we made, you have to be happy with the result. In the end we came close to winning it, but that would have been unjust," Ajax coach Frank de Boer said.
PSG took the lead just as the two sides were settling into the game. An error from Nicolai Boilesen, who let a long ball slip away from him, enabled Lucas to square a pass to Zlatan Ibrahimovic and although the visiting captain's shot was blocked, Cavani tapped home the loose ball.
Brazilian Lucas was at the heart of the French champions' creativity but was also the main culprit in their profligacy.
He stole the ball straight after the break but instead of passing to the unmarked Ibrahimovic he went alone and shanked his shot horribly wide.
Ibrahimovic missed from an angle with time to set his sights and then thundered a low free kick past the Ajax wall but straight at goalkeeper Jasper Cillessen.
Lucas set up Gregory van der Wiel, one of three former Ajax players in the Parisian line-up, as he overlapped down the right wing just after the hour mark but his effort was again easy for Cillessen to stop.
Lucas's floated free kick was then met with some power by the head of Marquinhos in the 67th minute but he was off target as PSG failed to exploit the gulf in class between the sides.
Ajax substitute Anwar El Ghazi might have got a red card just minutes after coming on as he kicked out at PSG full back Maxwell off the ball but instead was shown a lenient yellow by German referee Wolfgang Stark.
The hosts finally broke the shackles and after a rare attack they were awarded a free kick when Thulani Serero was brought down on the edge of the box by Marquinhos.
Schone flighted the ball past the deceived Salvatore Sirigu and three minutes later the Denmark international might have clinched an unlikely win but rattled the upright with another sweetly-struck free kick.
"We played a very strong team. We did well but we must do better. I had a golden chance early in the second half to make it 2-0 and I made a mistake with it," a rueful Lucas said.
(Reporting by Mark Gleeson; Editing by Steve Tongue)
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