The Browns announced the news and noted they plan to update his playing status in "a couple weeks."
Ian Rapoport of the NFL Network pointed out the ailment usually sidelines players for between four and six weeks, though individual timetables for a full recovery vary.
Garrett was unable to finish Wednesday's practice session after suffering the injury.
The 21-year-old Texas A&M product was a dominant force at the collegiate level when healthy. He racked up 141 total tackles, 31 sacks, seven forced fumbles and an interception across 34 games with the Aggies.
His 8.5 sacks last season were the lowest total in three years, though. The dip in production was at least in part attributed to a high-ankle sprain.
Patrick Maks of the Browns' official website passed along comments Garrett made about the ailment, which he said A&M offered to let him heal by sitting out for a while, but he declined.
"[It was] one of the worst foot injuries I've had, and I fractured a growth plate when I was in high school in a weightlifting accident," he said. "That didn't hang on for so long. It was a thing, it healed up and it went right back to working, but that ankle sprain hung on for a while."
Garrett also dealt with a lateral foot sprain following a minicamp practice in June.
Looking ahead, Cleveland is likely to take an extremely conservative path with his recovery. The Browns are looking at another year near the bottom of the standings as part of their rebuilding process, so there's no reason to rush him back into the lineup, barring an unexpectedly strong start to the season.
Nate Orchard and Carl Nassib are likely to split time to fill the void opposite Emmanuel Ogbah at defensive end until Garrett is cleared for his NFL debut.