Browns coach Romeo Crennel didn't provide specifics of Smith's injury but indicated the 6-foot-4, 310-pounder would be sidelined for an extended period.
"He is not in the best of shape," Crennel said. "For a big man, an Achilles is a tough thing to deal with."
The Browns, who fell to 0-2 with a 10-6 loss, are already playing without two key defensive players. Linebacker Antwan Peek is out for the season with a knee injury, and safety Sean Jones will be sidelined for several more weeks after undergoing arthroscopic knee surgery last week.
Smith started all 16 games last season with the Browns, who signed him as a free agent in 2007.