The clearance fell to Ben Chilwell who raced into the area to net from 12 yards as the keeper scrambled back.
The lively Son set up Kane for a 25th-minute opener and made it 2-0 with a first-time shot on the turn in the 36th minute as the visitors dominated the opening half with some crisp passing.
Kane then fired home from 20 meters late on for 5-1, before adding a fourth with another low effort from the edge of the box past Leicester goalie Kasper Schmeichel.
It was the kind of predatory performance that Pochettino believes further enhances Kane's reputation as one of the world's best finishers. "Then try to add more quality, maybe, if we are capable of finding this type of player that can help us for the next season to play better, to be better and try to compete better in different periods of the season". We are building a very exciting project and there is no reason for players like Harry to leave.
While Tottenham are still battling at the top of the table the Foxes have endured a different season, but a win tonight could push Craig Shakespeare's side as high as eighth depending on how many goals they score against the north London club.
Alli was the catalyst, taking possession on the edge of the area and impudently bypassing the Leicester defence with a sublime chipped pass to Son, whose half-volley lacked power but still had enough accuracy to beat Schmeichel.
Leicester's joy was short-lived when Spurs restored their two-goal advantage two minutes later through Kane's close range header.
"I don't know, you'll have to ask the owners about that", he said.