Uruguay striker Cavani converted from the spot midway through the second period after Kenny Lala's first half penalty had given Strasbourg a surprise lead.
PSG have drawn back-to-back league games, but they extended their unbeaten run to 16 matches this season and sit 14 points clear of second-placed Lille in the table.
Eric Maxim Choupo-Moting, Thilo Kehrer and Adrien Rabiot all threatened with early headers for the reigning champions before Thomas Meunier blazed off target from in front of goal in the 16th minute.
Kevin Zohi had Strasbourg's first attempt on goal in the 21st minute, but his shot was blocked and PSG resumed their offensive.
Julian Draxler's shot was held by Strasbourg goalkeeper Matz Sels, Cavani headed wide and Choupo-Moting's header struck a defender.
But it was Strasbourg who took the lead, courtesy of a handball incident that was referred to the video assistant referee.
PSG defender Kehrer was adjudged to have handled Lionel Carole's cross and after a four minute delay while the video was reviewed, Lala converted his spot kick into the bottom corner.
Kehrer tried to make amends early in the second half, but his header was saved by Sels and moments later Kylian Mbappe crashed a shot from outside the box just wide.
Moussa Diaby also went close from inside the area, but with 19 minutes remaining Cavani hauled PSG level when he coolly converted from the penalty spot after Lala's foul on Mbappe.
PSG went in search of the winner, but Strasbourg, 4-1 winners at Rennes last weekend, refused to wilt under the pressure.
Sels saved well from another Kehrer header and denied Choupo-Moting, whose goalbound effort was heading for the top corner, while Mbappe headed Dani Alves' cross just wide.
Strasbourg threatened to snatch victory in the closing stages when Adrien Thomasson forced Alphonse Areola into a fine save.