The Lakers reportedly applied for the exception Sept. 9.
Cousins tore his left ACL during a private workout last month. He had signed a one-year contract in July for just above the veteran's minimum.
An arrest warrant remains outstanding for the veteran NBA center in Mobile, Alabama, on a misdemeanor domestic violence charge.
The disabled player exception allows a team over the salary cap to acquire a replacement for a player who's out for a given season. The maximum salary of the replacement is either 50% of the injured player's salary, or the non-taxpayer mid-level exception, whichever is less.
The Lakers are currently $14 million over the cap.