A first goal for the club from Davinson Sanchez and a fine strike from Christian Eriksen were enough to give Mauricio Pochettino's side the win at Wembley.
Jamie Vardy, who missed a penalty with his first touch as a substitute, halved the deficit with 14 minutes to go but Son Heung-min struck a third on the break to seal the three points in injury-time.
Kasper Schmeichel blocked a Danny Rose strike after a good run from the left-back, but Leicester looked the more threatening of the sides in the opening half-hour, with James Maddison twice dragging shots wide from good positions.
It was Spurs who struck first after 33 minutes, though. Kieran Trippier sent a corner to Eriksen on the edge of the box, and his cross was headed in by Sanchez, who looked very close to being offside.
Demarai Gray headed a great chance straight at Hugo Lloris as Leicester threatened again early in the second half, with Claude Puel turning to Vardy on the bench before the hour-mark, but the striker saw his penalty comfortably saved by Lloris after Jan Vertonghen was judged to have fouled Maddison.
Three minutes later, Spurs had their second. Ricardo Pereira - who conceded possession in the build-up to Marcus Rashford's goal at the King Power Stadium last week - gave away the ball near his own box, allowing Fernando Llorente to set up Eriksen to fire beyond Schmeichel from 20 yards out.
Harvey Barnes produced another tame finish when sent through on goal by Vardy, before Pereira made amends for his earlier error by sending in a low cross that Vardy flicked past Lloris.
Any threat of a comeback was killed by Son in injury-time, though, as he raced clear on the break into Leicester's half of the field before slotting the ball beyond Schmeichel.
