England striker Kane cancelled out Jamie Vardy's early opener, but goals from Riyad Mahrez and Kelechi Iheanacho had the Foxes 3-1 ahead two minutes after the break.
Lamela inspired the fightback by twice finishing from Kyle Walker-Peters assists, and it was the Argentine's strike that forced an own goal from Christian Fuchs to put Spurs in front by the hour mark.
Vardy's expertly taken second appeared to have earned Leicester a remarkable point, but England team-mate Kane had the last word to seal a scarcely believable three points with his 11th Premier League goal against Leicester - more than any other opponent he has faced.
The dramatic victory means Spurs, who were already assured of a Champions League berth next season, finish third ahead of Liverpool, but Kane's unlikely pursuit of Mohamed Salah in the race for the Golden Boot fell short despite him scoring 30 Premier League goals in a season for the first time.
It remains to be seen if boss Mauricio Pochettino and key centre-back Toby Alderweireld, who have both been the subject of speculation surrounding their futures, will remain with Spurs when they leave the national stadium to play at their new ground on the site of the old White Hart Lane next season.
Claude Puel's own tenure at Leicester is also likely to face further scrutiny, the Foxes finishing the season ninth after ending the league campaign with just one win in seven – albeit that coming against Arsenal in midweek.
Spurs were caught cold in just the fourth minute when Vardy was allowed a free run to glance a header from Mahrez's teasing left-wing free-kick into the far corner.
The response was swift, though. Lucas Moura caught Danny Simpson dawdling in possession and Kane picked up the loose ball to race free and place a left-foot shot past Eldin Jakupovic.
A typically blood-and-thunder final-day contest continued in the 16th minute. Iheanacho wormed his way into the area, Adrien Silva prodded the ball towards Vardy and, although he was denied, Mahrez was on hand to smash home from 10 yards.
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.
The excitement continued two minutes after the break when Iheanacho held off Victor Wanyama's challenge, strode forward and rifled an unstoppable 25-yard effort into the top-right corner.
Leicester's lead was at two for just two minutes. Spurs' passing was one-touch and slick, leading to Walker-Peters drilling a ball across goal for Lamela to tap in.
By the 52nd minute Spurs were level. Rose's low cross from the left was controlled by Lucas and he teed up Lamela, whose shot ricocheted off Harry Maguire and Fuchs before nestling in the goal.
Incredibly Spurs were ahead by the hour. A poor defensive touch by Gray was pounced on by Walker-Peters, who once again kept his cool to pick out Lamela for the simple finish.
1995 – @SpursOfficial have finished as the highest placed London club in a top-flight season for the first time since 1994-95, when they finished 7th above six other sides from the capital. Boast. pic.twitter.com/Qp73cxxyvO— OptaJoe (@OptaJoe) May 13, 2018
The drama was not finished there. Shortly after Danny Rose was denied a penalty following a challenge from Silva, Mahrez – who had just missed a great chance – slipped in Vardy to smash home Leicester's equaliser for his 20th league goal of the campaign.
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.
Key Opta stats:
- Tottenham's 5-4 victory over Leicester equals the most goals scored in a competitive club match at Wembley Stadium (Chelsea 5-4 Man City in the 1986 Full Members Cup Final).
- This is the eighth occasion Tottenham have come from two goals down to win a Premier League game and the first since March 2014 against Southampton (3-2).
- In contrast, the Foxes have lost after leading by two goals in the top-flight for the first time since October 2003 (3-4 v Wolves).
- Harry Kane has scored more goals in all competitions versus Leicester City than against any other side in his career (13).
- Harry Kane is the first Spurs player to score 40+ goals in a season in all competitions since Clive Allen in 1986-87 (49).
- Kane has reached 30 Premier League goals for the first time in his career – the first Englishman to do so since Kevin Phillips in 1999-2000 (also, 30).
- Riyad Mahrez has assisted seven of Jamie Vardy's Premier League goals this season – no player has assisted another on more occasion during this campaign.
- Since his Premier League debut in August 2014, Jamie Vardy has scored 26 goals against the big six sides – six more than any other player during this period (Sergio Aguero, 20).
- Indeed, the England hitman has scored 11 goals in his last 12 Premier League games against the big six – finding the net in nine of those games.
- Vardy has been on the losing side in nine Premier League games this season when he has scored, extending the record in a single campaign in the competition.