Brown scored on a 1-yard run in the second quarter and a 4-yard run in the fourth.
Ricky Williams ran for a 1-yard score and totaled 85 yards rushing for Miami (1-3).
Miami also scored on a 23-yard interception return by Vontae Davis.
Trent Edwards, who was 14-of-26 passing for 192 yards, threw a 3-yard touchdown pass to Josh Reed for the Bills (1-3).
Buffalo's Terrell Owens caught three passes for 60 yards.
Susan Sarandon 'very excited' about daughter's pregnancy
Scarlett Johansson steps out with fiance after pregnancy reveal