The 19-year-old agreed a five-year contract with the north London club as he made a quick return to the Premier League following Burnley's relegation last season.
Tottenham moved for Odobert after the teenager's encouraging debut campaign in England.
He made 34 appearances and scored five goals for Burnley after joining from Troyes, becoming the Clarets' youngest Premier League goalscorer when he netted against Chelsea in October.
Odobert swaps the Clarets' promotion bid for Tottenham's attempt to finish in the Premier League's top four after they came fifth last term.
"We are delighted to announce that we have reached an agreement for the permanent transfer of Wilson Odobert from Burnley, subject to formalities," Tottenham said in a statement.
"The 19-year-old has agreed a deal that will run until 2029 and will wear shirt number 28."
Odobert could make his Tottenham debut in their Premier League opener at Leicester on Monday.
