The Jets (8-5) maintained their playoff hopes thanks in part to a career-high 129 yards on 24 carries by Shonn Greene.
New York got 181 yards through the air from Sanchez while extending its winning streak to three games.
Kansas City (5-8) was held to 21 rushing yards in the loss.
Tyler Palko managed 195 passing yards with one touchdown and an interception for the Chiefs.
Celebrity Families of 2014 [PHOTOS]