Moore racked up 244 aerial yards and Reggie Bush contributed 92 yards on the ground for the Dolphins (1-7), who logged their first win since Dec. 10, 2010, against the New York Jets.
Miami shined on defense, stopping the Chiefs four times on fourth-down plays.
Kansas City (4-4) was led by Matt Cassel, who passed for 253 yards on 20-of-39 throws.
The Chiefs saw their four-game winning streak snapped.
Notable deaths of 2014 [PHOTOS]