The Cypriot champions had their moments in Nicosia but three goals from Kane in a 28-minute period either side of half time ensured that Spurs - as well as Real Madrid - opened up a six-point gap at the top of the group.
Within two minutes of kickoff, Spurs could have been easily ahead through Kane, but the in-form Englishman dragged a low effort wide of the post from 20 yards out.
Son Heung-min then had a close-range volley well blocked by a defender but while Spurs dominated the opening quarter-of-an-hour, the Premier League side failed to take advantage and APOEL slowly but surely took control.
Roland Sallai was the first to squander a chance with the Hungarian heading wide from 12 yards, before Igor De Camargo drilled a left-footed effort against the crossbar after being played in on the left-hand side of the penalty area.
Soon after the half-hour mark, Carlao sent a free header wide from a corner and Davinson Sanchez was fortunate to avoid conceding an own goal after having the ball struck against him by goalkeeper Hugo Lloris.
Spurs head coach Mauricio Pochettino would have taken reaching the break on level terms but six minutes before half time, the North Londoners went in front through Kane, who slid the ball into the net after running onto a through-ball from Toby Alderweireld.
Within 30 seconds of the restart, Son blazed the ball across the face of goal after being teed up by Kane, but for the following 15 minutes, APOEL bossed the game.
Pochettino opted to make a change to his formation with Fernando Llorente joining Kane in attack after replacing Aurier and while Spurs looked equally vulnerable at the back, they soon put the result beyond any doubt.
Kane's second came courtesy of being picked out on the edge of the penalty area by Moussa Sissoko, with the striker taking his time before drilling a shot into the bottom corner of Boy Waterman's net.
Five minutes later, the hat-trick was confirmed when Kieran Tripper found the run of the club's senior marksman, who glanced a header into the corner from no more than eight yards out.
Son went close to a fourth as Spurs threatened to run away with the game but to APOEL's credit, they continued to fight until the end of the contest as the hosts searched for the consolation goal that their played deserved.
Spurs were able to coast through the remaining minutes of the game knowing that they head into their back-to-back showdowns with Real with the perfect record, and victory at the Bernabeu or Wembley Stadium will leave the club on the brink of qualification for the last 16.