On the sort of inhospitable evening that Stoke's stadium became famous for in the Premier League, Spurs boss Jose Mourinho paid the now-Championship team the compliment of fielding a strong side from the off including goal-machine Kane.
Chasing a first major trophy since winning the League Cup in 2008, the Premier League side totally dominated the first half and took a deserved lead when Gareth Bale flicked in a cute 22nd minute header from a Harry Winks cross.
It was Bale's third goal since returning to Tottenham on loan from Real Madrid.
The hosts lacked their famously passionate crowd at the Bet365 stadium, but the chill wind and pre-match rain bore out the old cliche about the toughest test in football being a "cold, wet night at Stoke."
And indeed, as those few present to watch huddled in their woolies, Stoke brought scowls to Mourinho's face in the second half with an equaliser by Jordan Thompson, who beat Hugo Lloris from close after Jacob Brown's pass in the 53rd minute.
Tottenham weathered the mini-storm, however, to restore their lead with a fine low shot from outside the area by Ben Davies in the 70th minute before Kane, inevitably, blasted into the top of the net from close range in the 81st minute.
"We're really happy. We knew that a cold, wet windy Wednesday night at Stoke was going to be tough," man-of-the-match Winks said afterwards.
"The boys were brilliant. It's a tough place to come."
Spurs were top of the Premier League earlier this month but they have dropped to sixth after collecting only a point from their last three matches. Mourinho has already won the League Cup four times with Chelsea and Manchester United and is now two wins away from a fifth triumph in the competition.
"We deserved it," he said after beating Stoke. "Now we have two tough games to come."
At Stoke, Mourinho started with out-of-favour Dele Alli, but he was substituted after Stoke's 37-year-old debut goalkeeper Andy Lonergan thwarted him twice with smart saves. It was Lonergan's first competitive game since turning out for third tier League One team Rochdale in 2019.
Stoke manager Michael O'Neill said Stoke were too timid in the first half, but he was not displeased overall.
"We expected Spurs to come with a strong team -- maybe not as strong as they started" he chuckled.
"Spurs showed their class. They showed real quality ... But we gave a decent account of ourselves."