Both teams will now play in the Europa League in the next season edition of the second division of Europe's competition.
Jamie Vardy converted two penalties to give Leicester 1-0 and 2-1 leads and, with Chelsea losing at Aston Villa, that would have been enough for Leicester to claim the fourth spot.
But a 76th-minute own goal by goalkeeper Kasper Schmeichel gifted Spurs an equaliser and Gareth Bale, possibly in his last game for Tottenham on loan, rubbed salt into Leicester's wounds with two well-taken late goals.
Just like the last term when a Champions League qualifying spot looked to be theirs for the taking, this season's FA Cup winners Leicester finished in fifth place and will have to content themselves with a Europa League campaign.
Victory left Tottenham in seventh and they will enter the inaugural Europa Conference League.