The Uruguayan had toiled all night and appeared to be carrying a knock but was on hand to give the French side an undeserved win that kept them top of Group F on seven points.
Devoid of any cohesion and rhythm going forward, PSG were unable to create a clear goalscoring opportunity all night against a well-drilled and organised APOEL side until Cavani managed to bundle the ball home from close range.
The Cypriot side, who are bottom of the group with one point, will feel they created the better chances during the match to merit a win, but were left to curse their luck as a lack of quality and two goal-line clearances from David Luiz kept them at bay.
APOEL started the brighter of the two sides and Cillian Sheridan forced Salvatore Sirigu into a good save in the third minute when he parried the Irishman's shot around the post following good work down the left by Gustavo Manduca and George Efrem.
Both teams set out their stalls very early with APOEL happy to concede possession and territory to their French opponents and keep 11 men behind the ball at times.
The French club struggled to break down a well-organised APOEL rearguard and it took until the 20th minute for them to record a shot on target through Javier Pastore, firing straight at Urko Pardo in the APOEL goal.
The Cypriots grew into the game as the first half wore on and PSG struggled to penetrate their well-marshalled backline.
Both teams spurned good opportunities towards the end of the half when Maxwell failed to control a Marco Verratti throughball, which would have put him clean through on goal, while Manduca's shot from the edge of the area for APOEL was deflected wide.
After the break, Cavani forced Pardo to tip his header over the bar from Lucas's cross and the Brazilian winger almost snuck in moments later but for a last-ditch tackle from his compatriot Carlao.
APOEL came close to opening the scoring in the 55th minute when Manduca's run and cross down the left found a wide open Tiago Gomes, but Luiz was on hand to clear off the line.
Luiz came to PSG's rescue again in the eight minutes later with another last-gasp clearance after Sheridan had managed to get a touch on the ball just before Sirigu.
APOEL were becoming more and more dangerous as the game went on while PSG lacked the spark needed to threaten the Cypriot team's goal. Cavani, however, came to the French side's rescue, after Carlao failed to clear.
