Vizcaino, 28, was placed on the 10-day injured list April 14 with shoulder inflammation.
He entered the season as one of the Braves' primary ninth-inning options and recorded one save over four appearances with a 2.25 ERA.
For his career, Vizcaino has a 3.01 ERA, 1.29 WHIP, and 10.10 K/9 over 194 1/3 innings. His 50 saves since 2015 are second among Braves relievers in that time to Jim Johnson's 51.
Left-hander A.J. Minter will likely get the bulk of the save opportunities with Vizcaino sidelined.
Atlanta has been one of the teams linked to free-agent closer Craig Kimbrel.