Skipper Billy Bremner led Don Revie's side to victory over Juventus.
Leeds beat Sarpsborg, Dynamo Dresden, Sparta Prague, Vitoria Setubal and then Liverpool in the semi final.
The first leg of the final against Juventus in Turin was abandoned in the 51st minute due to heavy rain and a waterlogged pitch.
The rearranged game finished 2-2 with Roberto Bettega and Fabio Capello scoring for the hosts and Paul Madeley and Mick Bates netting for Leeds.
At Elland Road, Clarke put Leeds ahead and while Pietro Anastasi levelled Leeds claimed the trophy on away goals after a 3-3 aggregate draw.
They had also won the competition in 1968 after beating Ferencvaros 1-0 on aggregate having been runners-up to Dinamo Zagreb in 1967.
Their 1971 victory was the last Fairs Cup final, as thereafter it came under the governance of UEFA and was replaced by the UEFA Cup.