Spurs are returning to their rebuilt White Hart Lane stadium next season after playing this term at the national stadium and braces from Kane and Lamela and a Christian Fuchs own goal gave them the win that sealed third place.
Leicester looked out of sight with a three-one lead but somehow contrived to let Spurs back into this game and they ended up on the wrong side of a nine-goal thriller with Kane scoring the vital goal - although his brace wasn't enough for the Golden Boot.
Kane's second goal of the game came in the 76th minute and Gascoigne produced a brilliant celebration when the camera panned to him in the stands.
The north Londoners, who were already assured of Champions League football next season, secured their third successive top-three Premier League finish by edging a insane nine-goal thriller against Leicester.
BC-SOC-Tottenham-Leicester, 438 Spurs leaves Wembley with 5-4 win vs Leicester, clinches 3rdAP Photo LON826, LON830, LON839, LON838, LON821, LON824, LON825, LON801, LON803, LON802Eds: APNewsNow.
Leicester ought to have been further in front on 26 minutes.
The response was swift, though.More news: City reach 100 points as Gabriel Jesus strikes late
Leicester were back in charge with 16 minutes on the clock and the goal owed much to an incision from Kelechi Iheanacho.
Spurs sought another response, but they had Hugo Lloris to thank for keeping them in the game after fine counter-attacking work from Iheanacho and Vardy saw Demarai Gray go through one-on-one. Dier's clearing header was weak and the Leicester forward held off Wanyama - no mean feat - before unfurling a precision left-footed drive that picked out the top corner. Spurs' passing was one-touch and slick, leading to Walker-Peters drilling a ball across goal for Lamela to tap in. Harry Maguire then had a shot which he scuffed just wide as the insane game was end-to-end.
Incredibly Spurs were ahead by the hour.
The drama was not finished there.
Leicester did not let their heads slip after the equaliser and found themselves regaining the lead just ten minutes later, when Riyad Mahrez found the bottom corner with a lovely finish from the edge of the box.
But the majority of supporters inside Wembley again rose to their feet when Kane bent a sublime effort from just inside the area into the right side of the goal to put the gloss on a memorable match.