However, Eden Hazard restored Chelsea's lead for a third time with 15 minutes remaining before Nemanja Matic capped off the scoring with a stunning fourth, setting up a showdown with either Arsenal or Manchester City in next month's final.
The Blues had seen their lead over Spurs at the top of the Premier League table cut from 10 points to four over the course of the month, and with one eye perhaps on the title race Antonio Conte left the likes of Hazard and Diego Costa on the bench.
Spurs, on the other hand, were close to full strength as they looked to improve their disappointing record at Wembley this season, but their quest to do so got off to the worst possible start when they fell behind after less than five minutes.
Toby Alderweireld's foul on Pedro earned him an early yellow card, but there was worse punishment to come for the Belgian as Willian subsequently buried the free kick into the corner on goalkeeper Hugo Lloris's side.
The goal marked a bright start to the match for the Blues, but Spurs soon settled into it and were level by the 18th minute when Kane stooped to meet Christian Eriksen's delivery at the near post and flicked a clever backward header into the far corner.
Despite the absorbing nature of the match, clear chances were relatively hard to come by in the first half and it was not until nine minutes before the interval that either side next came close when Eric Dier flicked Jan Vertonghen's deep cross narrowly past the post.
Spurs were just about on top as Thibaut Courtois made a routine save to deny Eriksen's long-range effort moments later, but it was Chelsea who took the lead against the run of play with just three minutes remaining in the half.
It came in controversial fashion as Victor Moses went down under the challenge of Son Heung-min inside the penalty area and, while the South Korean was ill-judged to go to ground, replays suggested that Moses was on his way down before the tackle arrived.
It was easy to see why referee Martin Atkinson pointed to the spot nonetheless, and Willian - close to joining Spurs before he signed for Chelsea in 2013 - made no mistake by sending Lloris the wrong way from 12 yards for his second of the match.
That sent Chelsea into the break with a narrow lead, but it took Spurs just seven minutes after the restart to peg their London rivals back once again as Alli produced an emphatic finish to fire Eriksen's sublime ball past Courtois.
Conte called for the cavalry on the hour mark in an effort to regain control of the match, with Costa and Hazard replacing Michy Batshuayi and goalscorer Willian for the final half-hour.
Spurs may have been glad to see the back of the Brazilian, but they were given an untimely reminder of Chelsea's strength in depth when Hazard gave the Blues the lead for the third time with just 15 minutes remaining.
A corner was only cleared as far as the Belgian on the edge of the area, and Spurs could not get out in time to prevent him from taking a touch and drilling a low strike into the bottom corner.
Spurs quickly set off in search of a third equaliser, but their hopes of ending their semi-final hoodoo were blasted away in emphatic fashion when Matic capped off the scoring with a thunderous 25-yard effort that left Lloris no chance in the Tottenham goal.
Four goals was harsh on a Tottenham side who had more than held their own for the most part, but it was almost worse in the closing stages when Marcos Alonso and Costa both wasted clear chances to pour salt into the wound for their London rivals.
Spurs were close to pulling one goal back in the third minute of stoppage time when Kane's drilled free kick squirmed under the body of Courtois, but the backspin on the ball saw it slow down enough for the Chelsea keeper to recover before it crept over the line.
In all likelihood it would have been too little too late anyway as Spurs fell to a record seventh consecutive FA Cup semi-final defeat - a run which stretches back to the last time they lifted the trophy in 1991.
Chelsea have now made nine finals in that time, winning the trophy on six occasions since Tottenham's last triumph, and they will have the chance to draw level with Spurs' tally of eight competition wins when they face either Man City or Arsenal in the final.
Both teams must first turn their attention back to their ongoing title race, though, with Chelsea hosting Southampton on Tuesday and Spurs taking on Crystal Palace a day later.