Spurs were fortunate to remain on level terms through the first 39 minutes until Kane broke the deadlock with a cool finish under goalkeeper Boy Waterman.
Read below to see how the game unfolded in Nicosia.
6.45pmHello, and welcome to Sports Mole's live coverage of the match in the Champions League between APOEL and Tottenham Hotspur.
6.47pmWhen Spurs came out on top against Borussia Dortmund on Matchday One, everything appeared to be rosy as far as the start of their European campaign was concerned but on Monday, Mauricio Pochettino revealed the extent of the club's selection problems ahead of their trip to Cyprus. With that in mind, who does the Argentine go for this evening? Let's find out.
6.49pmTOTTENHAM HOTSPUR XI: Lloris, Trippier, Alderweireld, Sanchez, Davies, Aurier, Dier, Winks, Sissoko, Son, Kane.
6.50pmTOTTENHAM HOTSPUR SUBSTITUTES: Vorm, Foyth, Walker-Peters, Oakley-Boothe, Georgiou, Nkoudou, Llorente
6.52pmAPOEL XI: Waterman, Vouros, Rueda, Carlao, Roberto Lago, Morais, Vinicius, Sallai, Zahid, Aloneftis, De Camargo
6.53pmAPOEL SUBSTITUTES: Gudino, Alexandrou, Merkis, Farias, Makris, Pote, Antoniou
6.56pmThe main absentee from the Spurs starting lineup is Christian Eriksen, who misses out through illness, while Jan Vertonghen is not involved after his late dismissal against Dortmund. Dele Alli must also sit out the second of three European matches after his sending off in the Europa League last season, so it leaves Pochettino having to tinker with his first XI.
6.58pmThe most notable addition to the team is Kieran Tripper, mainly because the England international looks set to work in tandem with fellow right-sided player Serge Aurier. It remains to be seen which of the pair will operate further forward, but chances are that it will be Aurier.
7.01pmIn midfield, Harry Winks has been introduced for what will be just his second start of the season - the other coming in last week's EFL Cup triumph over Barnsley. The 21-year-old will presumably link up with the likes of Eric Dier and Moussa Sissoko, but there are a number of formations which Pochettino could go with tonight.
7.03pmSon Heung-min also comes back into contention after spending the early stages of the campaign in and out of the team. The South Korean star got on the scoresheet on Matchday One, however, so he has already made a positive contribution to Spurs' efforts to progress through to the group stages.
7.06pmAs far as the bench is concerned, it is not necessarily down to the bare bones, but it isn't far off. Fernando Llorente can probably expect some action later in the game, while Tashan Oakley-Boothe and Anthony Georgiou will hope to gain some first-team experience if Spurs hold a sizeable lead.
7.09pmLet's take a look at APOEL now with the standout name being Nuno Morais, who has gone on to forge a career as an APOEL stalwart since representing Chelsea on five occasions between 2004 and 2007. He is partnered in midfield by the experienced Vinicius, who gained plenty of playing time in Portugal before moving to APOEL in 2013.
7.12pmFurther forward, the APOEL frontline is headed by Igor de Camargo, a Brazilian-born striker who has won nine caps for Belgium. The 34-year-old has made over 250 appearances for the likes of Standard Liege and Borussia Monchengladbach so he has plenty of pedigree.
7.15pmAPOEL also have 16 years worth of experience between the sticks, too, with Boy Waterman taking responsibility for trying to keep Spurs at bay this evening. The 33-year-old has previously represented the Netherlands Under-21 side on 13 occasions and was part of the squad which won the Under-21 European Championship in 2007.
7.18pmWe should also mentioned manager Girogos Donis, who spent part of his playing career with Blackburn Rovers, Sheffield United and Huddersfield Town. The 47-year-old is in his second spell with the Cypriot giants, having previously completed the league and cup double in 2014.
7.21pmFORM! Spurs head into this match sitting in fourth place in the Premier League table, but the North Londoners would be higher had their home form matched their results on the road. While Pochettino's side have failed to earn success at Wembley Stadium in the top flight, three successive away victories have been registered, with eight goals being scored and just two being conceded.
7.24pmWhat will be most encouraging for Pochettino and his side will be their ability to score goals on the road. The eight goals that we have just mentioned have come at Newcastle United, Everton and West Ham United respectively. In the case of the trips to Goodison Park and the London Stadium, it could have been a lot more, too. Spurs have only looked vulnerable when Serge Aurier was sent off during the final quarter of Saturday's win over West Ham.
7.28pmIf we are to take a look at the Cypriot Premier League, we can see that APOEL have endured a somewhat indifferent start to the campaign. Donis's men currently sit in 11th place with four points from three games, with their last two contests seeing them fail to net against either Anorthosis or Aris. Life is certainly not perfect in Nicosia.
7.31pmHowever, to get to this stage of the competition, APOEL have had to come through three qualifying ties, with a 2-0 aggregate success over Slavia Prague proving enough to earn another appearance among the elite on the continent. It is also worth pointing out that five clean sheets have been kept in six matches. Maybe Harry Kane will not have it all his own way this evening.
7.34pmAPOEL are not too familiar with facing English opposition in Europe. Their last encounters came back in 2009 when they met Chelsea. A Nicolas Anelka goal proved enough for the Blues in Cyprus but in the final group game, a Nenad Mirosavljevic effort three minutes from time earned the minnows a surprise draw at Stmaford Bridge.
7.37pmSpurs last face Cypriot opposition during the 2014-15 Europa League, where they recorded a 5-1 aggregate win over AEL Limassol in the playoff round. The away leg saw Spurs struggle, with goals from Roberto Soldado and Kane eventually proving decisive in a 2-1 win.
7.40pmPREDICTION! Given Spurs' recent away record, we cannot go against them for this match. APOEL are likely to provide a stern test but if Spurs can make the breakthrough, it is difficult to see APOEL having enough in reserve to prevent the Premier League side from getting the job done. Spurs to win 3-1.
7.42pmThe two teams have made their way out at the GSP Stadium. Kickoff is coming right up.
1 minAPOEL get us underway.
2 minStraightaway, we can see that Spurs are playing with three at the back, with Ben Davies taking the place of Vertonghen. Aurier is at right wing-back and Tripper is on the opposite flank.
2 minSHOT! It takes less than two minutes for Kane to get a shot on goal, with the forward dragging an effort from 20 yards wide of the post. Strong start from the English outfit.
4 minCHANCE! APOEL are all over the place at the back. A rapid break from Spurs ends with Aurier crossing from the right. The ball is only half-cleared to Son, who sees a volley deflected wide of the post from close range.
7 minAPOEL have started to settle now - to a certain extent, anyway. That said, every time Spurs have the ball, the home side are straight onto the defensive. They understand the threat which is being offered by the visitors.
9 minYou already get the impression that Spurs can make this an easy night's work if they want to. There is a gulf in class between the two clubs. Spurs have had at least 75% of possession.
11 minSon is given time and space to get a shot away from around 20 yards out after a neat flick from Kane, but the South Korean is too complacent and that allows a defender to make a block.
14 minHuge credit to the home supporters. They have had very little to shout about so far but that has not stopped them raising the decibel levels. Spurs are not intimidated, however. If the visitors can net a goal, it could be a stroll in the park.
16 minNot much more to update you on. Spurs have delivered a low cross into the area, but it was comfortably cleared.
17 minCHANCE! Big chance for APOEL, but it is poor defending from Spurs. An inswinging cross finds the head of Sallai but his free header from around 12 yards goes wide of Lloris's post.
18 minCHANCE! Now it is Spurs' turn to waste a headed chance. Tripper cuts inside onto his right foot before finding Son all on his own in the middle. However, the ball is marginally too high for the attacker and he directs his header over.
19 minWOODWORK! Blimey. APOEL have started to create the better chances now. De Camargo is played in on the left-hand side of the penalty area and the Belgian international drills his shot off the crossbar before seeing the ball bounce clear. Massive let-off for Spurs.
22 minLATEST! There has been a goal in the other match in this group, and it is Real Madrid who have taken the lead against Borussia Dortmund. Gareth Bale has got on the scoresheet.
24 minThose chances for APOEL have only stoked up the atmosphere. From being outclassed in the opening quarter-of-an-hour, APOEL are now on top. Spurs need to regain control of the tempo.
26 minWe have had nine shots on goal in these early stages but we are yet to see an effort on target. The 10-yard strike from De Camagro which hit Lloris's bar is the closest we have come.
28 minSpurs are normally ruthless in front of goal but they are taking a touch or second too much each time they are in and around the penalty area.
30 minWe have reached the half-hour mark and while Spurs will be content to be on level terms, Pochettino will not be pleased with the performance. They are making heavy weather of this contest now.
32 minSpurs win a corner on the right and the inswinging delivery is intended for Dier. However, the versatile Englishman is never going to score with his near-post effort.
34 minThis is so poor from Aurier. The Ivorian initially did well to break forward to the by-line but his eventually delivery was appalling and too strong for Son. Kane looked to be the pass in the middle.
36 minCHANCE! Yet another massive chance for APOEL. Carlao peels away at the far post to meet a corner but his free header goes wide of the post from no more than eight yards. It should be 1-0.
37 minCHANCE! Lloris smashes a clearance into the body of Sanchez and the ball trickles all the way towards goal before going a yard wide of the post. The French stopper - and Spurs - are incredibly fortunate to still be level.
39 minGOAL! APOEL 0-1 Tottenham Hotspur 1 (Kane)
39 minSpurs have the breakthrough against the run of play. An outstanding through-ball from Alderweireld finds Kane, who eases off the defender before sliding the ball past the goalkeeper and into the net. It's a clinical finish from the in-form striker.
42 minIn the aftermath of the goal, Waterman went into the book for complaining about the lack of offside. The assistant got it spot-on - the right-back was playing Kane on.
44 minSpurs are just attempting to play down the clock until the break now. I don't blame them. They need to take some life out of this game, especially now that they are ahead.
45 minHALF TIME: APOEL 0-1 Tottenham Hotspur
8.35pmThe half-time whistle goes in Cyprus and Spurs, somehow, lead this match thanks to Kane's goal. Before that, APOEL squandered two clear opportunities, while Lloris was fortunate to avoid putting his team behind by firing the ball into Sanchez and seeing the ball trickle into the net. There is all to play for in the second half.
46 minSpurs get us back underway. There have been no changes for either side.
46 minCHANCE! Spurs could be two ahead within 30 seconds of the restart. Kane plays in Son down the left-hand side of the area but the attacker's low effort goes wide of the far post. In all fairness, he was attempting to score from an acute angle.
48 minSHOT! APOEL are straight back onto the offensive too with Sallai seeing a volley saved by Lloris. I wasn't expecting this to be that much of an entertaining game but it has been highly competitive.
51 minWe haven't seen much of Dier and Winks since the opening stages of the game. Spurs need one or both of the midfield pair to start to dictate play. They are spending too much time chasing the ball.
51 minBOOKING! Vinicius is the second APOEL player to go into the book for a crude challenge in the middle of the park.
54 minLATEST! There has been a second goal in Dortmund and Real now lead 2-0. Cristiano Ronaldo has got onto the scoresheet. As it stands, Spurs will be delighted with the current scores.
56 minCHANCE! Aloneftis is sent clear of the Spurs defence and he has the pace to hold off the attention of Aurier. He still has plenty to do but the winger forces a fine one-handed save out of Lloris.
57 minSUBSTITUTION! This is interesting from Pochettino. He has hooked Aurier and replaced him with Fernando Llorente, who is playing in behind Kane. I think Spurs have reverted to four at the back.
58 minLATEST! Dortmund have halved the deficit against Real Madrid. Pierre-Emerick Aubameyang has got the goal.
60 minThere are definitely more goals in this match. I'm just not so sure whether they will be for Spurs. Like in the first half, APOEL are the better team and they deserve to be at least level.
62 minGOAL! APOEL 0-2 Tottenham Hotspur (Kane)
62 minThat, ladies and gentleman, is called the commentator's curse. Sissoko breaks down the right and after biding his time, picks out Kane on the edge of the box. The striker is patient before firing a low effort into the bottom corner for his second of the evening. That could be that.
64 minSHOT! Sissoko must be really starting to impress Pochettino. The Frenchman gets into a similar position and with no support to his left, takes a shot from 20 yards, only to see Waterman parry the ball away.
67 minGOAL! APOEL 0-3 Tottenham Hotspur (Kane)
67 minClinical. Ruthless. Efficient. All of those words apply to Kane. Tripper's cross picks out the run of Kane who glances a header into the bottom corner from eight yards. Yet another treble. What a striker.
69 minSHOT! Spurs could run up a cricket score here. Son has all the time in the world to shoot on goal from the edge of the area but his attempt is deflected marginally wide of the post.
72 minIncidentally, it has been a perfect hat-trick for Kane. It hasn't been the perfect Spurs performance, but most definitely the result. They are going to have six points to their name ahead of back-to-back fixtures with Real Madrid.
75 minSUBSTITUTION! Pochettino makes a smart move in replacing Kane with Georges-Kevin N'Koudou.
76 minSHOT! To their credit, APOEL continue to push forward. Zahid has just let fly with a low strike from distance, but the ball was too close to Lloris.
79 minSHOT! Once again, Zahid gets another shot away after Spurs fail to deal with a cross, but his weak volley is deflected away for a corner. It's not going to be a foregone conclusion that Dortmund will collect six points against APOEL.
83 minWe have entered the stage of the game where it is starting to peter out. It's job done for Spurs, even though they have needed to survive some sticky moments.
84 minSUBSTITUTION! Pochettino takes the opportunity to introduce Anthony Georgiou, who is going to make his first-team debut for Spurs. Sissoko is the man to make way.
85 minBOOKING! Winks goes into the book for a needless foul on the edge of the box.
86 minLATEST! Real have made sure of the points in Dortmund, with Ronaldo making it 3-1 to the defending champions.
88 minThe former Walsall man Andreas Makris has been introduced and he should probably shoot after breaking clear of Davies. However, he opts for the cutback and it is behind its intended target. A waste.
90 minTherere will be three minutes of added-on time.
90+3 minFULL TIME: APOEL 0-3 Tottenham Hotspur
9.39pmThe full-time whistle goes in Nicosia and Spurs have been able to comfortably see out what turned out to be an impressive win in Cyrpus. The Premier League outfit certainly didn't have it all their own way but Kane's treble has ensured that the North Londoners are looking in great shape ahead of their meetings with Real Madrid.
9.40pmThat will be all from Sports Mole for this game. Thank you for joining us this evening and we hope to see you again tomorrow when the likes of Manchester United, Chelsea, Barcelona, Paris Saint-Germain and Bayern Munich will all be in action. Goodbye for now.