Washington scored on runs of 18 and 12 yards, while Nugent kicked field goals from 29, 40, 35 and 26 yards out.
Kellen Clemens was 15-for-24 passing with 236 yards, one touchdown and one interception for the Jets (3-9), who won on the road for the first time in six tries. Laveranues Coles had five catches for 69 yards for New York.
Thomas Jones also ran for a Jets score.
John Beck completed 23 of 39 passes for 177 yards and three interceptions for the Dolphins (0-12), who are on their way to becoming the first team since the 1976 Tampa Bay Buccaneers to go through an entire season winless.