Cozart was placed on the 10-day disabled list June 15 after initially being diagnosed with a shoulder strain, but additional imaging and clinical examination revealed a need for surgery.
Before he hit the shelf, the 2017 All-Star slashed .219/.296/.362 with five home runs and 18 RBI.
The Angels have yet to provide a timetable for Cozart's recovery.
In the meantime, Luis Valbuena should pick up additional reps at the hot corner. So far this season, the veteran infielder has hit .216 with nine home runs and 26 RBI across 229 plate appearances.
Valbuena has also posted a .965 fielding percentage in 337.1 innings at third in 2018.