The 19-year-old arrived at Turf Moor from Troyes 12 months ago and scored four times in 33 matches last term as the Clarets were relegated from the top flight.
Odobert became the club's youngest Premier League goalscorer when he struck in a 4-1 defeat against Chelsea last October.
He also netted against Crystal Palace and Fulham, earning rave reviews for his performances on the left flank and through the middle despite a tough season for the Clarets.
The French Under-21 international was on target in the 4-1 win at Luton Town on Monday night as Scott Parker made a winning start to his Burnley tenure.
He now returns to the top flight with Spurs, who are aiming to build on last season's fifth-place finish.
An official statement read: "The club would like to wish Wilson all the best and thank him for his efforts during his time at Turf Moor."
Odobert has put to pen over a deal running until 2029 with the London outfit and will wear the number 28 shirt.
Ange Postecoglou's side kick off the campaign with a trip to the King Power Stadium to take on newly-promoted Leicester City on Monday night.
Spurs have also brought in the likes of Dom Solanke and Archie Gray this summer from Bournemouth and Leeds respectively.