The England striker broke the deadlock in the 39th minute with a cool finish under goalkeeper Boy Waterman, whose side had been dangerous up until that point in the game.
APOEL threatened after the break but two goals from Kane in the space of five minutes ensured that Spurs recorded their second successive victory in Group H.
"It is a very proud night for myself and the team. I try and score every game - It is a bonus when I get three," Kane told BBC Sport after the game.
"We weren't pleased with the first half and a couple of chances could have gone the other way. We were more clinical and in the Champions League that is what you have got to be.
"We are missing a few players but the lads who stepped in were fantastic. We have a solid squad and you have to be ready. 3-0 away in the Champions League no matter who you play is a good result."
Tottenham's 100% start leaves them joint top of Group H with Real Madrid.