Texas announced the move before Thursday's game, two days before what would have been Klein's next scheduled start.
Alberto was hitting .303 with two home runs and 13 RBIs in 42 games at Triple-A, and at a .375 clip since May 2. He was looking to make his major league debut.
Klein was 1-0 with an 8.25 ERA in eight games, including two starts with the team. He started Monday in Cleveland.
The Rangers said their starter for Saturday against the Red Sox was to be determined.