Poyet has left Elland Road with the club's best wishes after Leeds agreed an undisclosed compensation package with Tottenham.
Leeds granted the London club permission to speak to their assistant manager yesterday morning following an official approach for the Uruguayan.
A Leeds statement read: "The club would like to thank Gus for his efforts over the past 12 months and wishes him every success in his new role. He leaves with the best wishes of everyone connected with the club."
Poyet arrived at Leeds in October 2006 with former Chelsea team-mate Dennis Wise.
The duo had forged a managerial partnership the previous summer at Swindon where they had guided the Robins to the top of Coca-Cola League Two.