The Uruguay international struck with just three minutes remaining in Cyprus to ensure that Laurent Blanc's men remain above Barcelona at the top of Champions League Group F
With talisman Zlatan Ibrahimovic still on the sidelines through injury, Cavani fired home in the 87th minute to keep PSG top of Group F.
Having beaten Barcelona 3-2 in their previous Champions League encounter, PSG will have expected to run out comfortable winners in Tuesday's clash in the Cypriot capital of Nicosia.
But they were frustrated for long periods by the hosts, who produced a stern defensive performance and threatened PSG sporadically in what was a largely uninspiring game.
Yet just when the contest appeared to be meandering to a draw, Uruguay international Cavani produced a determined solo effort to seal a dramatic victory.
The result puts Laurent Blanc's men on seven points, one ahead of Barca and five ahead of third-placed Ajax.
The visitors settled after that scare but were unable to craft any clear-cut goalscoring opportunities, despite enjoying an almost complete dominance of possession, with a frustrating first half encapsulated in the 26th minute as Edinson Cavani failed to make a clean connection with the ball following great work down the right from Lucas Moura.
Cavani then glanced a near-post corner into the side-netting and APOEL defender Marios Antoniades was taken off four minutes before half-time to be replaced by Champions League winner John Arne Riise.
APOEL goalkeeper Urko Pardo was forced into his first real save of the match in the opening minute of the second half as he tipped Cavani's header over the crossbar.
The hosts remained a threat going forward, though, and would have taken the lead if not for a superb piece of defending from Luiz.
Tiago Gomes' header looked destined for the goal, only for former Chelsea man Luiz to intervene with a magnificent goal-line clearance.
Luiz then came to rescue again to clear to ensure that Sheridan did not put APOEL ahead after he had lifted the ball over an onrushing Sirigu.
But for all APOEL's endeavour, they were denied what would have been a deserved point when Cavani turned brilliantly on the edge of the area from Jean-Christophe Bahebeck's knock-down before holding off two players and firing a low effort into the bottom right corner.