Marsh was educated at Woolwich Polytechnic and represented Greenwich as a Labour Party Member of Parliament between 1959 and 1971. He served in the second Wilson Government as the Minister of Power (1966-1968) and in the Cabinet as Minister of Transport (1968-1969).
Following his departure from the House of Commons, he served as Chairman of the British Railways Board (1971-1976). He received a Knighthood and later a life peerage as Baron Marsh, of Mannington in the County of Wiltshire in 1981. He now sits in the House of Lords as a Crossbench peer.