Newton's scoring pass came at the end of an 80-yard drive with time winding down.
He finished with 225 yards of total offense and ran for a score as the Panthers (8-3) remained one game behind NFC South-leading New Orleans.
The Dolphins (5-6), coming off a victory over San Diego last week, were held without a point in the second half.
Ryan Tannehill found Mike Wallace on a 53-yard score in the first quarter for Miami's only touchdown.
Notable deaths of 2014 [PHOTOS]