The former England midfielder, 39, is joined at Elland Road by assistant Gus Poyet after the pair linked up at former club Swindon Town.
A statement on the Leeds website said: "A compensation package has been agreed between the two clubs.
"Swindon Town have placed on record their thanks to Leeds United for the professional manner they have made their approaches."
Wise and Poyet will officially take up their posts today while goalkeeping coach Andrew Beasley has also joined the club's staff.
Wise's appointment has not been universally popular among Leeds fans, but club chairman Ken Bates defended the move.
"After much deliberation the unanimous choice was Dennis Wise and his assistant Gus Poyet," he said in his programme notes last night.
"I am aware that he may not be everybody's cup of tea but this is not a popularity contest.
"It is a question of the right man to resolve a serious situation."
Kevin Blackwell was sacked on September 20 with coach John Carver taking over in a caretaker role.
However, Leeds have since slipped to second from bottom in the Championship after a run of four straight defeats.
Carver left the club on Monday as Leeds agreed personal terms with Wise.
Wise joined Town in May following their relegation to League Two and they won their first six league games.
However, a clause in his and Poyet's contracts allowed them to speak to Premiership and Championship clubs, provided suitable compensation was agreed.