Mujica, who has a 3.68 ERA in 16 appearances for the A's since his May trade from the Red Sox, compiled 37 saves for the Cardinals in 2013, his only other stint as a full-time closer.
The right-hander will have several setup men behind him, including right-hander Fernando Rodriguez and left-handers Fernando Abad and Drew Pomeranz, who will also remain an option to pitch multiple innings.
Melvin said he thought about utilizing Pomeranz in the ninth inning "real briefly, but I like where he's at now," he said.
Pomeranz, who carried a 4.63 ERA in nine games as a starter, entered Tuesday's matchup against the Dodgers with a 1.40 ERA in 21 appearances out of the bullpen.
Right-hander R.J. Alvarez was called up from Triple-A Nashville to join the A's bullpen in time for the game, taking the roster spot of Clippard. Alvarez, who posted a 13.00 ERA in 10 appearances over his first two stints with the A's this year, had five saves with a 4.13 ERA in 21 games for the Sounds.