Brown racked up 65 yards on 13 carries for Miami (1-0), which also got 62 yards on the ground from Ricky Williams on 18 attempts.
Dolphins signal-caller Chad Henne went 21-of-34 through the air, amassing182 yards for the victors.
The Bills (0-1) were led by quarterback Trent Edwards, who tossed for 139 yards on 18-of-34 passing, connecting on a fourth-quarter touchdown.
Chan Gailey marked his debut as Buffalo's head coach with a defeat.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys