Defending champions PSG headed to Stade de Gerland knowing a win would see them leapfrog Lyon at the summit of the French top flight.
But Laurent Blanc's men appeared destined for defeat after Clinton N'Jie put Lyon ahead just after the half-hour mark.
PSG were continually thwarted by a string of fine saves from Lyon goalkeeper Anthony Lopes, who was eventually beaten from 12 yards by Ibrahimovic.
After seeing the Portuguese creep off his line to keep out his first effort, Ibrahimovic dispatched the retake to keep the capital club within two points of Hubert Fournier's side.
The visitors continued to dictate proceedings and would have deservedly gone ahead had David Luiz not nodded wide with the goal at his mercy after Cavani had flicked Thiago Motta's corner into his path.
PSG were made to pay for their profligacy in the 31st minute - N'Jie finishing off a well-worked team move.
Jordan Ferri whipped in a left-wing cross for N'Jie, who worked a superb one-two with Nabil Fekir before emphatically lashing the ball beyond Salvatore Sirigu.
Blanc's men will have been frustrated to go into the interval behind given their relative dominance, which became apparent again after the restart as Lopes continued to keep PSG at bay.
Cavani then failed to get enough purchase on a low cross from Blaise Matuidi and was thwarted by Lopes again shortly after when through on goal following Thiago Motta's excellent throughball.
PSG finally did break Lopes' resistance in the 69th minute, though.
Lindsay Rose conceded the spot-kick for hacking down Marco Verratti in the box, but Lopes looked to have saved the day with a diving save to his left from Ibrahimovic's initial penalty.
A last-ditch challenge from Marquinhos prevented Fekir from restoring Lyon's advantage, as PSG settled for a result that at least prevents the leaders from taking control of an enthralling title race.
