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 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.
Spurs' title challenge trundles on
Spurs are back to within five points of leaders Liverpool and you have to consider them to have a shout of a title shot, with 12 games still to play.
This was another outing where they rode their luck at times, especially when Vardy's penalty miss and a series of poor finishes from Maddison and Barnes are taken into account, but Pochettino's side simply keep finding a way to get the points.
Leicester's 12th defeat of the season keeps them 12th in the table, and Puel's future is likely to remain under scrutiny.
Sanchez solid for shaky Spurs
Pochettino's defence did not always look convincing, making Sanchez's performance at the heart of the back four particularly key. He also took his goal well with a clever header.
Maddison not ready to match Eriksen
Maddison has been excellent for Leicester this season, but he was outshone by Eriksen, the man he has been tipped to replace at Spurs. He wasted a number of good opportunities with poor shots or final passes.
