Bush's performance -- the first 200-yard rushing game by a Dolphin since Ricky Williams gained 216 in 2002 -- gave interim head Coach Todd Bowles a victory in his first game.
Bowles took over Monday from the fired Tony Sparano.
Matt Moore passed for a pair of scores en route to 217 yards as the Dolphins (5-9) won for the fifth time in seven games following an 0-7 start.
The reeling Bills, meanwhile, lost for the seventh straight game.
Ryan Fitzpatrick went 31-of-47 through the air for 316 yards with two touchdowns but was intercepted three times for Buffalo (5-9).
Senate Democrats to pull all-nighter on climate change
Susan Sarandon 'very excited' about daughter's pregnancy