The Foxes, who had not scored in their last three league games, seemed to lack confidence early on and Spurs dominated the opening 15 minutes.
Christian Eriksen and Dele Alli threatened with efforts from distance, while Toby Alderweireld went close with a header from Erik Lamela's corner that an off-balance Kasper Schmeichel just about tipped over.
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After surviving the early onslaught, Leicester began to grow into the game. Shinji Okazaki and N'Golo Kante both missed the target from inside the area, while Danny Drinkwater's curling effort from distance was parried out by Hugo Lloris.
Chances were few and far between in the remainder of the first half, with Eriksen's dipping drive from distance that Danish teammate Schmeichel palmed away the only moment of note.
Shortly after the hour mark Kane was in more familiar territory at the other end of the field and he was inches from giving Spurs the lead, receiving a pass from Lamela before his dinked effort was parried by Schmeichel into the turf and off the crossbar.
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Both sides had excellent openings with 15 minutes remaining as first Lamela blazed over from Ben Davies's cutback before Riyad Mahrez, Okazaki and Leonardo Ulloa were all denied by goalline blocks within the space of 10 seconds in the Spurs area.
Minutes later Schmeichel sprawled to his left to deny Alli and then Leicester stunned the home side in the 83rd minute as Huth powered home a header after being left unmarked at the back post to meet Christian Fuchs's corner.
The Foxes then defended well to see out the win and collect three points which move them level on points with Arsenal, who were held to a 3-3 draw at Liverpool, at the top of the table. Spurs stay fourth and drop seven points behind the pace.