The deadly front man came up with a crucial left-footed opener six minutes before the break to take the sting out of APOEL's spirited charge before drilling in Moussa Sissoko's pass with his right foot on 62 minutes, then completing his treble with a header five minutes later.
That made it seven goals in five career Champions League appearances for our home-grown hitman, who has now scored in four consecutive ties in the competition dating back to Monaco away last November.
APOEL did hit the woodwork through Igor de Camargo in the first period while Hugo Lloris had to make a couple of smart saves after the break, but we were always in control of the Group H tie.
There was a nice moment six minutes from time, too, as young winger Anthony Georgiou, who is eligible to play for Cyprus, came on for his senior competitive debut in place of Sissoko.
It means we're on six points out of six in the group ahead of back-to-back matches with holders Real Madrid next month.
With a raft of players unavailable due to injuries, suspensions and, in the case of Christian Eriksen, illness, Mauricio Pochettino was forced to shuffle his pack. Ben Davies moved infield to operate as the left of our three centre-backs in place of the suspended Jan Vertonghen - a position he's familiar with for Wales - while Serge Aurier started at right wing-back, with Kieran Trippier switching to the left. Harry Winks, impressive in recent outings, came in to start in central midfield alongside Eric Dier, with Heung-Min Son and Sissoko behind Kane.
CHANCES: EARLY SHOTS AND HEADERS - 2-18mins
The opening exchanges were by the book - our Lilywhite shirts in firm control of the game without really testing home goalkeeper Boy Waterman. Kane dragged wide from range after a long ball forward less than two minutes in, while Son's close range shot was blocked for a corner soon afterwards after the ball fortuitously fell to him. Both sides then traded aerial chances - Roland Sallai glancing wide for APOEL while Son got under Trippier's in-swinger to head over the top for us.
WOODWORK: DE CAMARGO NEARLY BREAKS THE DEADLOCK - 19mins
We had a let-off on 19 minutes when Aurier got the ball stuck under his feet and APOEL took advantage, working the ball to De Camargo who struck the crossbar with a fierce strike from the left angle.
CHANCES: APOEL CLOSE TWICE IN TWO MINUTES - 36-37mins
De Camargo's chance really lifted the hosts, along with the noise from the home fans, and they might have counted themselves unlucky not to have taken the lead when Carlão peeled away from Toby Alderweireld to plant a header wide from a 36th-minute corner, before Lloris skewed a clearance straight at Davinson Sanchez but saw the rebound trickle wide a minute later.
Above: Heung-Min Son celebrates with Harry Kane and Moussa Sissoko after Harry's first goal.
GOAL: APOEL 0-1 SPURS - HARRY KANE - 39mins
Who else to beat the off-side trap and break the deadlock? Six minutes before half-time, Alderweireld sent a long ball forward and Praxitelis Vouros played Kane on-side, allowing the deadly striker to fire past Waterman, who was booked for his protestations afterwards.
SAVES: LLORIS CALLED INTO ACTION - 48-56mins
We made a lightning-fast start to the second half with Son zipping a shot across goal and wide after just 26 seconds, but APOEL also came out firing and it needed a good save from Lloris to his left to deny Sallai on 48 minutes. A short while later, Stathis Aloneftis raced onto a through ball between Aurier and Alderweireld and pulled the trigger from the left angle, but Lloris tipped away.
GOAL: APOEL 0-2 SPURS - HARRY KANE - 62mins
The introduction of Fernando Llorente for Aurier saw us switch to a back four and straight away we were causing APOEL more problems. Goal number two arrived shortly after the hour mark when Sissoko did well on the right side before pulling the ball across for Kane to drill into the bottom corner of the net from just inside the area.
GOAL: APOEL 0-3 SPURS - HARRY KANE - 67mins
Kane's hat-trick was completed on 67 minutes. It all started with a confident burst forward from Davies, Kane then took over, laying the ball out wide on the right. Sissoko let it run for Trippier, who delivered an inch-perfect cross for Kane to run onto and nod home from inside the box. Son blasted just off target two minutes later, but we had the game wrapped up.
MAURICIO POCHETTINO SAID
"It was an important result, three points, fantastic for us. It was a game we must win, we won and we were competitive. For us it’s fantastic and we can look forward now, six points and it’s true there is still a long way to go but to start with two victories is so important for us and now it’s important to be focused on the game against Huddersfield on Saturday because that will be tough as well. It was so important to be clinical. Not only do you have to play well but you have to be clinical in front of goal as well. It was another fantastic night for Harry, amazing. A goal with the left, the right and the head, a perfect hat-trick. I’m so happy for him and the team because the effort was fantastic."
MOUSSA SISSOKO SAID
"To come here and win 3-0 is not easy. We knew it was going to be tough. APOEL started well and hit the crossbar but we stayed focused and won 3-0 so it was the perfect night. In the Champions League every game is going to be tough. You have to play every game like a final. That’s what we did against Dortmund and now APOEL and we’ve got six points, the perfect start."
APOEL (4-3-3): Waterman, Vouros, Rueda, Carlao, Roberto Lago, Morais (c, Alexandrou 83), Sallai (Makris 62), Vinicius, Zahid, De Camargo, Aloneftis (Farias 75). Substitutes (not used): Gudino, Merkis, Pote, Antoniou.
Spurs (3-4-2-1): Lloris (c), Alderweireld, Sanchez, Davies, Aurier (Llorente 57), Dier, Winks, Trippier, Sissoko (Georgiou 84), Son, Kane (Nkoudou 74). Substitutes (not used): Vorm, Walker-Peters, Foyth, Oakley-Boothe.
Goals: Spurs - Kane 39, 62, 67.
Yellow cards: APOEL - Waterman 39, Vinicius 51; Spurs - Winks 85.
Referee: Pavel Královec.
MAN OF THE MATCH
Hat-trick hero Harry Kane took Man of the Match honours from our supporters on social media with a whopping 83 per cent of the votes.
Attentions now turn back to the Premier League and a trip to one of the clubs APOEL manager Giorgos Donis used to play for, Huddersfield Town. Newly-promoted from the Championship, Huddersfield have won two, drawn three and lost one of their opening top-flight fixtures this season. Kick-off is at 12.30pm.