With the Premier League side missing the creative spark of both Christian Eriksen and Dele Alli, it was down to the English striker to pick up the slack, and despite a spirited effort from the home side, Spurs ran out easy winners.
Apoel started shakily and could have been behind on more than one occasion, Heung-Min Son missing the best chance for Spurs when he headed over unmarked from five yards out.
The Cypriots slowly grew into the game, striking the bar and coming inches from opening the scoring themselves, following a calamitous clearance from Tottenham goakeeper Hugo Lloris just after the half-hour mark.
But it was Spurs who went ahead in the 38th minute when Kane finished coolly after he was left unmarked, 18 yards out. He was expertly found by Toby Alderweireld, placing the ball past the on-rushing Boy Waterman, despite appeals for offside from the home side.
Lloris was forced into his first meaningful save in the 47th minute when he was equal to Hungarian forward Roland Sallai’s volley from just inside the penalty area.
The French number one was tested again shortly after, this time by the lively Stathis Aloneftis, who was put through by Nuno Morais down the left. His high shot was tipped over by Lloris.
Spurs doubled their lead and Kane got his second in the 62nd minute when Moussa Sissoko found the forward on the edge of the box and the Englishman made no mistake with his right foot.
Kane completed his 'perfect' hat-trick five minutes later by heading in a Kieran Trippier cross from the right to seal the win for Spurs.
Tottenham now have six points, along with Real Madrid who beat Borussia Dortmund 3-1 in Germany. Both APOEL and Dortmund remain without any points.
Wasteful Liverpool held to draw
Liverpool's stuttering early-season form continued as they wasted a string of chances and needed a first-half equaliser from Philippe Coutinho to secure a 1-1 draw with Spartak Moscow.
Juergen Klopp's side fell behind and maintained their recent habit of conceding soft goals when goalkeeper Loris Karius was slow to react to a curling free kick from 25m by Spartak's Fernando, which sailed into the net in the 23rd minute.
The visitors were back on level terms, however, eight minutes later when Sadio Mane, returning after a three-match domestic suspension, combined brilliantly with Coutinho, before the Brazilian fired the ball above Spartak goalkeeper Artem Rebrov.
With draws in their first two Champions League fixtures, Liverpool and Spartak have two points, and sit two points behind Group E leaders Sevilla, who beat Maribor 3-0.
It will go down as another disappointing result for Liverpool, whose Premier League form this season has also been hampered by costly defensive mistakes and an inability to kill off matches when on top.
Against Spartak, they squandered several opportunities either side of halftime and failed to capitalise on defensive mistakes from the hosts, whose domestic form has been poor this season to leave them ninth in the 16-team Russian top flight.
In the dying minutes of the first half, Mohamed Salah fired the ball across an empty net after Liverpool managed to break four on two after a defensive blunder by Spartak.
After making a string of key saves, Spartak keeper Rebrov was taken off the pitch in the 67th minute following a collision with Salah.
Yet substitute Aleksandr Selikhov was up to the task, denying Salah in the fifth minute of stoppage time with a point-blank save from a header, after Liverpool striker Daniel Sturridge had come off the bench to blast another superb chance over from close range.
