Jose Mourinho's side plays Everton in the fifth round to maintain its challenge in four competitions.
Gareth Bale scored only his fourth goal since returning to Tottenham in leading a comeback to beat Wycombe 4-1 in the FA Cup on Monday.
Playing his first full match of the season, Bale netted before halftime to cancel out Fred Onyedinma's shock opener for the second-tier side.
With extra time looming, Harry Winks put Tottenham in front with a curling strike in the 86th minute then Tanguy Ndombele netted twice.
Jose Mourinho's side plays Everton in the fifth round to maintain its challenge in four competitions.
Mourinho named a strong team and this was a chance for Bale to show why he should be in the team more often while back on loan from Real Madrid.
He wasted a chance after five minutes by heading wide and dragged an effort wide from another good position on the edge of the area.
Those misses proved costly as Wycombe took the lead in the 25th minute.
Uche Ikpeazu got the better of Tottenham defender Toby Alderweireld down the right and his cross was not cleared by Davinson Sanchez, who could only divert it into the path of Onyedinma to fire home.
Spurs went for a quick response and the chances continued to rack up. Japhet Tanganga headed over from a corner immediately after, Bale hit the side-netting with another good chance, and Moussa Sissoko’s deflected effort from the edge of the area cannoned into the crossbar.
Mourinho may have been fearing the worst when he saw Tanganga head another golden chance against the crossbar, but his side finally levelled in first-half stoppage time.
Lucas Moura found space on the left and Bale broke the offside trap to hook home from close range.
That ought to have been the platform for Spurs to go on and win the game comfortably in the second half but a raft of missed chances caused unnecessary stress.
Bale was again wasteful after surging into the area, Erik Lamela could not get a proper connection on his shot from a good position and fired straight at Ryan Alsopp, and then Moura poked wide after finding space on the edge of the area.
Mourinho was not prepared to risk anything so he sent on Kane and Son Heung-min, but they too were equally wasteful.
Kane saw his shot blocked by Alsopp after miscontrolling, and Son flashed an effort wide after a trademark dart into the area.
But Spurs won it with a two-goal salvo in the space of 72 seconds.
After Kane spurned another chance at the far post, the ball fell to Winks 20 yards out and the midfielder sent a curling effort into the top corner.
Immediately from kickoff, Spurs got in down the left, Son’s shot was deflected into the path of Ndombele and the Frenchman diverted into the bottom corner. He scored a second in stoppage time when he superbly cut inside and fired in at the near post.
