The England international stole the headlines once again as Spurs drew level on points with fourth-placed Liverpool, overtaking club legend Teddy Sheringham to stand out on his own as Tottenham's most prolific player of the Premier League era.
Son Heung-min had earlier opened the scoring on his way to a man-of-the-match display, and Christian Eriksen capped off the rout with the pick of the bunch just nine minutes from time as Spurs made it 11 consecutive league games without defeat at Wembley.
Everton handed a debut to new £27m man Cenk Tosun following his move from Besiktas, but it was the home side who were the first to threaten after 15 minutes when Son flicked Eriksen's wicked free kick narrowly over the crossbar with Jordan Pickford rooted to the spot.
The visitors responded with their first sight of goal just a minute later when Tosun steered the ball into the path of Wayne Rooney, but the former England captain dragged his subsequent effort wide of the target.
Rooney did have the ball in the back of the net midway through the half when he again linked up with his new strike partner, diverting Tosun's header into the roof of the net from close range only to see his celebrations cut short by the offside flag.
That scare seemed to spark Tottenham into life, and Kane in particular began to threaten moments later with three efforts in quick succession which all needed to be saved by Pickford.
It was Son who broke the deadlock in the 26th minute, though, as Serge Aurier took advantage of some lax defending before dragging an attempted shot into the path of the South Korean, who tapped the ball into an empty net from close range.
The goal saw Son net for a fifth consecutive home Premier League game, equalling Tottenham's club record which was originally set by Jermain Defoe between November and December 2004.
Not even the prolific Kane has managed that feat, but the England international did add another record to his growing list of accolades just two minutes into the second half when he netted his 97th Premier League goal for Tottenham, drawing level with Sheringham.
Son was again involved down the left channel as he burst forward before dragging a shot across goal which Kane swept home in similar fashion to Tottenham's opener, although there was a question of offside against the league's top scorer.
Suddenly Tottenham looked dangerous every time they came forward, and they should have had a third five minutes later when Dele Alli exchanged passes with Son on his way into the box, only to fire his finish into the side-netting with only the keeper to beat.
Kane then drew another save from Pickford before the influential Son came within a whisker of deservedly doubling his personal tally when he collected Kane's crossfield pass, cut inside past two defenders and then fired a low strike against the post with the keeper beaten.
Spurs were rampant, though, and just a minute later they had their third goal as Kane surpassed Sheringham as Tottenham's highest-ever Premier League goalscorer, bringing up number 98 with another close-range finish, although this one was scuffed over Pickford from Eric Dier's drilled low delivery.
The goal was also Kane's 20th of the season, seeing him join an elite club of Thierry Henry and Alan Shearer as the only players in Premier League history to have hit that tally in four consecutive campaigns.
Having hit eight hat-tricks in 2017 Kane was left looking for his first of 2018 in the final 20 minutes, but Spurs should have added a fourth from a different source when Son cut inside from the left flank only to drive his finish straight at Pickford, who also recovered to deny Alli from the follow-up.
All four of Tottenham's attackers were involved when the fourth did arrive shortly afterwards, though, with Kane feeding the ball into Son, who flicked it on for Alli to back-heel into the path of Eriksen, and the Dane produced an emphatic first-time finish to cap off a devastating team move.
The fourth goal did not flatter a Spurs side who now sit level on points and goal difference with Liverpool ahead of the Reds' Anfield showdown with league leaders Manchester City on Sunday.
Everton, meanwhile, slump to a fourth consecutive defeat across all competitions and a sixth game in a row without victory, during which time they have only managed two goals.
Indeed, the Toffees did not register a single shot on target this evening and have now lost their opening three games of a calendar year for the first time since 1960.
TOTTENHAM (4-2-3-1): Lloris; Aurier, Sanchez, Vertonghen, Davies; Dier (Wanyama, 74'), Dembele (Sissoko, 82'), Eriksen (Lamela, 87'), Alli, Son; Kane
EVERTON (4-2-3-1): Pickford; Kenny, Holgate, Jagielka, Martina; McCarthy (Schneiderlin, 72'), Gueye, Bolasie (Lennon, 57'), Rooney, Sigurdsson; Tosun (Calvert-Lewin, 62')