The Dons and Morton are on course for their second meeting at Hampden in the semi-finals of the Scottish League Cup.
Aberdeen won the first, 2-1, on December 1, 1979 – but it was a close fought affair.
Mark McGhee, who had scored for Morton when they beat the Dons in a League Cup tie five years before, put the Pittodrie men ahead with a header after 14 minutes.
When Gordon Strachan made it 2-0, from the penalty spot, just before the break it looked as if Aberdeen were going to stroll it.
Morton had other ideas, though.
Their midfield maestro Andy Ritchie rattled the woodwork before beating Dons keeper Bobby Clark from the penalty spot in the 81st minute.
Morton were also upset referee Willie Waddell disallowed a screamer of an effort from Neil Orr.
The Dons lost 3-0 to Dundee United at Dens Park in the replayed final.
The first match at Hampden had ended 0-0.