Gareth Bale broke the deadlock on the 22nd-minute mark for the away side in what was a dominant first-half display.
Jordan Thomson levelled up the scores nine minutes after the interval, but a stunning hit by Davies and a smashed finish by Kane ensured that Spurs booked themselves a spot in the last four of the competition.
Story of the game
The visitors started the encounter the brighter and they tested debutant Andy Lonergan for the first time when Matt Doherty slipped in Kane down the right-hand side, but his powerful strike was well parried away to safety by the 37-year-old.
Then at the midway point of the first half, Spurs took a deserved lead when a Harry Winks cross found Bale unmarked inside the area and he guided a header into the bottom right-hand corner.
On the 36th minute mark, Jose Mourinho's side had a chance to double their advantage when Kane played in Dele Alli down the left-hand side, but his low shot was blocked by the legs of Lonergan.
Two minutes before the break, Lonergan denied Alli again when the midfielders curling effort from range was well tipped around the post by the experienced goalkeeper.
Three minutes after the interval, Alli had another golden opportunity to grab his side a second when Kane found him inside the area, but he couldn't provide the telling touch to guide the ball past Lonergan.
Then, totally against the run of play after 54 minutes had been played, the Potters levelled when a Jordan Brown cross from the right was met by Thompson at the back post and he directed the ball into the bottom left-hand corner.
After this leveller, the hosts looked the more likely to go ahead, but with 20 minutes remaining Davies retained Spurs lead when his low driven shot from range arrowed off the post and into the back of the net to hand the away side the advantage.
Half time substitute Heung Min Son thought he had sealed a spot in the semi-finals for his side when he lobbed Lonergan inside the area, but he was adjudged to be offside after being fed by Kane.
Although only six minutes later the win was secured when Thompson sloppily gave the ball away allowing Kane to get into the box and he had made jo mistake as he managed to smash the ball past Lonergan and into the roof of the net.
Substitute Steven Fletcher had a great chance to make it an interesting last couple of minutes, but he failed to connect cleanly to a cross from the right as Spurs, in the end, managed to ease through to the Carabao Cup semi-finals.
Spurs quality eventually showed
Despite making seven changes from the defeat at home to Leicester City at the weekend, Mourinho still fielded a very strong Spurs side with the likes of Kane, Bale and Ali making the starting eleven.
It was a clash that as expected the away side dominated particularly in the first half as Stoke sat in deep and they just looked to soak up the pressure and close down the spaces in dangerous areas.
However, the failure for the away side to tuck away the chances gave the chance to equalise and even though Spurs struggled to show too much creativity in the second period, the quality of the players under their ranks was too much for Stoke to handle.
This was as late pieces of brilliance from in-form Kane and Son and slightly surprisingly from Davies proved to be enough to force a Spurs victory over the line.
Positives to take for the Potters
Despite coming away from the cup tie with defeat there are plenty of positives that Michael O'Neil can take from his side's showing.
Coming into the game they were huge underdogs and for a large portion of the game, they kept the attacking threat that Spurs to a minimum.
Then especially in the second half when breaking forward they looked dangerous and lively and when the scores were level at 1-1, it was Stoke that looked the more likely to take the lead, but unfortunately for them, it just was not meant to be.
As has been the case on several occasions so far this season, Kane was magnificent in particular in his hold up when coming to receive the ball from deep.
On many occasions, he decided to come in deep to allow room further up the field for the likes of Ali which through Kane's play, he did get on a number of occasions.
Kane also managed to get on the scoresheet for the 16th time in all competitions this season with an excellent finish past Lonergan.
Netting past the Potters is something that has become a regular occurrence as well for Kane, as tonight's strike was his 10th goal against Stoke in his last six outings against them.