The 28-year-old Mujica is 0-3 with a 4.38 ERA and two saves in 41 games and has held right-handed hitters to a .211 batting average. Last year he was 9-6 with a 2.96 ERA.
The 23-year-old Cox was batting .254 with nine homers and 30 RBIs at Triple-A Memphis. He was the 25th overall pick in the 2010 amateur draft out of Arkansas and jumped from Class A Palm Beach to Double-A Springfield last season.
The Cardinals will announce a corresponding roster move after Mujica arrives in Colorado for a series against the Rockies that began Tuesday night.
On Twitter, Cox said Marlins fans were already welcoming him.
"here I come #miamimarlins #stoked looking forward to my bright future it's been a good run with the cardinals," he wrote.
Information from The Associated Press was used in this report.