Son Heung-min opened the scoring with a close-range finish after 26 minutes, but it wasn't until the second half that Spurs really pulled away from their out-of-form visitors.
Kane opened his account just two minutes after the interval and then added a second to overtake Teddy Sheringham on 98 Premier League goals for the club, before Christian Eriksen capped off the scoring with the pick of the bunch nine minutes from time.
Find out how all of the action unfolded courtesy of Sports Mole's minute-by-minute updates below.
These two sides come into this match in contrasting form, and Spurs in particular need a win to avoid losing further ground in the top-four race this season. Everton have hit a slump following a good start to life under Sam Allardyce, so they too will be desperate for a return to winning ways here.
We'll have a close look at both teams a little later, but first let's check out the team news...
Well, we will get to Everton and a debut for their newest arrival in a short while, but let's start with the hosts today as Tottenham recall a host of big guns after resting them in the FA Cup win over Wimbledon last weekend.
Mauricio Pochettino has made eight changes to that team, with captain Hugo Lloris and the attacking midfield trio of Dele Alli, Christian Eriksen and Son Heung-min all amongst those to return to the side. All seven players on the bench today started last weekend, but drop back out here.
Indeed, a goal for Son today would see him become only the second Tottenham player to score in five consecutive Premier League home games - a feat not even Harry Kane has achieved yet. Indeed, the only player to do that for Spurs in the past was Jermain Defoe between November and December 2004.
The England international has a good record against Everton too, scoring twice in both of his last two league appearances against the Toffees. If Spurs can keep hold of him then there is no reason why Kane couldn't set an unreachable tally for the club.
There is only one change from Tottenham's most recent Premier League outing, though, with Dembele replacing Sissoko in a holding role alongside Eric Dier.
Sam Allardyce has regularly mentioned that scoring goals has been his side's biggest problem so far this season, and he has wasted no time in throwing Tosun into the mix this afternoon. He is straight in from the start, replacing Calvert-Lewin in the team from the FA Cup defeat to Liverpool.
Tosun also hit the 20-goal mark in the league last season to help Besiktas retain their title, and since the beginning of last term he has found the back of the net 38 times in 71 games across all competitions.
As an added bonus, Tosun also has experience of scoring a decisive goal against Tottenham, netting the only one of the game in a 1-0 Europa League victory in 2014 which saw Besiktas beat Spurs to top spot in Group C.
Behind them is the slightly surprising inclusion of Idrissa Gueye, who was expected to miss this game with an ongoing hamstring problem. However, he has recovered in time to feature from the start, replacing Schneiderlin in the XI today.
Captain Phil Jagielka gets another start alongside Holgate - who was of course involved in a controversial incident with Roberto Firmino last time out - while Kenny is on the right and Martina continues on the left. Pickford, as ever, is between the sticks.
Spurs stalled a bit during the latter months of 2017 to slip out of the Champions League places and, while they have not yet managed to work their way back into the top four, their run of 16 points from their last seven league games is second to only Man City in that time.
On the flip side, a draw or defeat could see them end the weekend down in sixth place should Arsenal - who are currently two points behind their North London neighbours - also win.
That run - which stretches back to November - includes seven victories and the only defeat in that time came against Manchester City, since when they have won four and lost none of their last five games.
The Hammers are, incidentally, one of only two visiting teams to have beaten Spurs at Wembley across all competitions so far this term, doing so with a 3-2 triumph in the EFL Cup.
Tottenham's only two home defeats this term both came in London derbies - against Chelsea and West Ham - and after getting off to a shaky start at the national stadium they now finally appear to be making it feel like home.
Spurs only picked up two points from their first three league home games this season - dropping more points in that time than they did throughout the entirety of their final season at White Hart Lane, but since then they have picked up 20 points from the 24 on offer.
Spurs still only boast the sixth-best home record in the division, but they are slowly turning Wembley into a fortress like they did with White Hart Lane.
It is a tactic which has served Allardyce well countless times throughout his career, but the first murmurs of his 'negative' style not being suited to Everton have just begun to be heard amongst the Toffees fans following their recent slump in form.
Everton come into this match winless in their last five outings across all competitions, losing their last three at the hands of Bournemouth, Manchester United and Liverpool, but perhaps an even more telling statistic is that they have scored only two goals in that time.
Since it was announced that he would be becoming manager Everton went on an eight-game unbeaten streak to soar up the table, from the relegation zone to the top half, so their position now compared to what it was before Allardyce is certainly a major improvement.
The Toffees have picked up just two points from the last 12 on offer in the league, scoring only one goal in that time, and another defeat here would make it three league losses on the bounce for the first time since September and only the second time all season.
Across all competitions Everton have won two, drawn two and lost two of their six away games since Allardyce's appointment, which is a mixed record but a significantly better one than their run of just one win in the 20 away games which preceded his arrival.
A 1-0 victory at Newcastle exactly a month ago is Everton's only away league win in their last 19 attempts, with nine draws and nine defeats in that time too. Only Southampton, West Brom and Stoke have picked up fewer points on the road this season.
Indeed, this season they are yet to beat any of the teams who began the weekend ahead of them in the table, drawing three and losing seven of their 10 such outings.
Allardyce rarely makes it easy for big teams when looking to grind out a result, but the form book clearly points to a Tottenham win and, with home advantage also in their favour, I am backing Pochettino's side to come out on top this evening.
SPORTS MOLE SAYS: Tottenham 2-0 Everton
Indeed, in the Premier League era Spurs boast 24 victories over Everton, which is more than they have managed against any other team.
Tottenham also won the corresponding fixture last season by scoring three goals including a Kane brace, although Everton mustered two in reply on that occasion with Alli, Lukaku and Enner Valencia also on the scoresheet.
The Toffees' last three visits to Wembley have also ended in defeat, all by a 2-1 scoreline against Chelsea in the 2009 FA Cup final, Liverpool in the 2012 FA Cup semi-final and Manchester United in the 2016 FA Cup semi-final.
TOTTENHAM STARTING XI: Lloris; Aurier, Sanchez, Vertonghen, Davies; Dier, Dembele, Eriksen, Son, Dele; Kane
EVERTON STARTING XI: Pickford; Kenny, Holgate, Jagielka, Martina; McCarthy, Gueye, Bolasie, Rooney, Sigurdsson; Tosun
Aurier is left in acres of space down the right channel, so much so that his first touch is poor but there are still no Everton players close to him when he lets fly from inside the box. Martina should come across but instead allows Aurier to produce an unchallenged shot, which he skews across goal. Son is there to tap in, though, and the replay proves that he was onside too.
Son scores in his fifth consecutive home game in the Premier League to put Spurs in front.— Sky Sports PL (@SkySportsPL) January 13, 2018
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Spurs have seen the majority of possession and had the better chances too, with Pickford by far the busier of the two keepers. Clear-cut openings have been few and far between, though.
It was poor defending from Everton as they left Aurier in acres of space down the right flank, and the full-back's dragged shot fell into the path of Son, who tapped home into the empty net from close range.
Harry Kane has been relatively quiet on the ball, although he did have a three-minute spell during which he drew three saves from Pickford. Pickford would have expected to make each of those saves, though, and aside from that it has been slim pickings for the England striker.
Rooney has come closest for the Toffees after latching on to Tosun's touch forward, but he could only drag his effort wide of the target with a left-footed strike.
Son is again heavily involved as he charges forward down the left flank and goes for goal himself, although in a similar way to his own goal he drags his shot and sees it tucked home by a teammate. This time it is Kane, who sweeps his effort into the back of the net for his 97th Premier League goal - drawing level with Teddy Sheringham in Tottenham's all-time scoring list.
This time it is Dier who drills a low cross into the box and Kane's movement sees him meet the ball ahead of the Everton defenders. His finish comes off his studs and goes into the ground before bouncing over Pickford and in. Not what Kane meant, but they all count!
Tottenham's top ten Premier League goalscorers of all time:— BBC Sport (@BBCSport) January 13, 2018
98 - Harry Kane
97 - Teddy Sheringham
91 - Jermain Defoe
91 - Robbie Keane
48 - Chris Armstrong
42 - Gareth Bale
36 - Steffen Iversen
36 - Christian Eriksen
34 - Darren Anderton
33 - Emmanuel Adebayor#THFC #TOTEVE pic.twitter.com/Fw4shHRV6v
Son feeds the ball in to Alli, who produces a back-heel into the path of Eriksen, who sweeps an emphatic first-time finish past the keeper. Every Tottenham attacker knew exactly where his teammate was then, and Everton simply couldn't cope.
It is a deserved win for Spurs too following a very impressive second-half showing, condemning Everton to a fourth consecutive defeat across all competitions.
Thank you very much for joining Sports Mole for today's late Premier League kickoff as Tottenham thrash Everton on an historic day for Harry Kane, whose brace sees him surpass Teddy Sheringham as the club's all-time leading scorer in the Premier League. I will leave you with our match report, and be sure to stick around for reaction too.
From me, though, it is goodbye for now!