Agreement was reached for the 25-year-old, who has played 13 times for England, late on transfer deadline day on Monday but the move hung in the balance as Villa were made to wait for the league's approval after the close of the window.
Cleverley, who came through United's academy system and has made 79 appearances for the 20-times English champions, made the move away from Old Trafford after finding playing opportunities limited under new manager Louis Van Gaal.
His departure follows the high-profile deadline-day exits of Javier Hernandez, who joined Real Madrid on a season-long loan, and Danny Welbeck, who signed for Arsenal in a deal reported to be worth 16 million pounds ($26.4 million).
It completed a busy transfer window for United as Dutchman Van Gaal, yet to win a competitive game since taking charge, seeks to return the club to winnings ways.
They spent nearly 150 million pounds, including the British record signing of Angel Di Maria from Real Madrid and Colombia striker Radamel Falcao, on loan from Monaco.
($1 = 0.6056 British Pounds)