The Ibrox club came from behind as John Greig claimed his first trophy as manager and denied Alex Ferguson's men the silverware.
Duncan Davidson scored for the Dons - who had Doug Rougvie sent off - but Alex MacDonald equalised and Colin Jackson headed in a last-minute winner.
Winning line-up
Peter McCloy
Sandy Jardine
Ally Dawson
Derek Johnstone
Colin Jackson
Alex MacDonald
Tommy McLean
Bobby Russell
Billy Urquhart
Gordon Smith
Davie Cooper
Substitutes
Alex Miller
Derek Parlane