The Uruguay striker levelled one minute into stoppage time with a superb 20-metre free kick after Neymar had been dismissed in the 87th minute for two yellow card offences in quick succession.
Neymar had earlier cancelled out Luiz Gustavo's opener before halftime but lost his cool in the closing stages after Florian Thauvin had put the home side ahead in the second half, threatening to inflict PSG's first defeat of the season.
Then Cavani, who has often been criticised for supposed lack of efficiency, netted his ninth league goal of the season to earn the draw that puts PSG on 26 points from 10 matches.
"It's an important goal because the game was almost finished but at least we salvaged a point," said Cavani.
Monaco are second, four points behind on 22 points after beating Caen 2-0 on Saturday.
Marseille are fifth on 18 points as they were leapfrogged into fourth on 19 by Olympique Lyonnais, who hammered Troyes 5-0 away with a Memphis Depay hat-trick.
In a boiling hot Velodrome, Marseille went ahead on 16 minutes when Gustavo beat Alphonse Areola with a 25-metre strike to give Marseille a deserved advantage after a solid start.
Cavani squandered an opportunity to equalise one minute later when he missed the target from close range from a Layvin Kurzawa cross.
Neymar levelled with his first opportunity in the 33rd as he collected a clever cross from Adrien Rabiot to beat Steve Mandanda with a left-footed shot.
PSG had the upper hand but the hosts defended well to preserve the score until halftime.
Kylian Mbappe was handed a yellow card for protesting after a Jordan Amavi handball in the box as PSG's frustration grew.
In the 78th minute, Clinton Njie, who came on for Konstantinos Mitroglu in the 65th, grabbed the ball from Rabiot on the right flank and Thauvin latched onto his cross to put the home side ahead.
Neymar was booked in the 85th minute for dissent and two minutes later reacted angrily after being fouled, only to be shown the red card. He will miss next weekend's home game against Nice.
In his absence, there was no debate about who would take the final free kick in injury time as Cavani preserved the leaders' unbeaten record.
(Reporting by Julien Pretot; Editing by Ian Chadband)
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