Holders Celtic will face the winners of the St Mirren versus Hearts replay in the semi-finals of the Scottish Cup.
Aberdeen take on Hibernian in the other all-Scottish Premier League tie after the draw at Fir Park following the Dons' 2-1 win over 10-man Motherwell.
Celtic had earlier overwhelmed 4-0 a Dundee United side who had also been reduced to 10 men for much of the game at Tannadice.
The ties at Hampden Park will be played on 14/15 April.
Hearts travel to St Mirren Park on Wednesday 21 March for their replay after the Buddies came from behind to draw 2-2 at Tynecastle on Saturday.
The draw avoided a possible Edinburgh derby, Hearts city rivals, Hibs, having seen off the last challenge from the First Division with a 2-0 win over cup specialists Ayr United.
Instead, Pat Fenlon's side will take on Craig Brown's Aberdeen, with the latter expressing his pleasure at avoiding the runaway SPL leaders and Cup holders.
"We're delighted to be in the draw," said the Dons manager. "All the teams are fairly even.
"Celtic are the best team I think and to avoid them is fortunate, but I'm not taking anything for granted.
"You see the way they played, we made heavy weather of the game today, so we need to improve against Hibs in the semi-final.
"But we were playing a very good Motherwell side with few injuries.
"I'm delighted for the Aberdeen fans and they deserve it.
"They've travelled everywhere and the support we have has been incredible"