Bush's scoring play put the Dolphins up 14-6 and Dan Carpenter added a 46-yard field goal later in the quarter to provide some insurance for Miami (2-3), which broke a two-game losing streak.
Ryan Tannehill completed 17-of-26 passes for 223 passing yards and Reshad Jones intercepted Bengals quarterback Andy Dalton at midfield to break up a last-chance drive by Cincinnati.
Dalton threw for 234 yards with a TD but was also picked off twice for the Bengals (3-2), whose three-game winning streak was snapped.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys