Derek McInnes' Dons, who have not reached the final of the Scottish Cup since 2000, came from behind - thanks to goals from Russell Anderson and Peter Pawlett - to knock holders Celtic out of the competition with a 2-1 win on Saturday at Parkhead.
Aberdeen, who last won the cup in 1990, remain on target to record a cup double for the first time since 1985-86 as they are due to face Inverness in the League Cup final on March 16.
Rangers were drawn at home against the lowest-ranked club, Albion Rovers. The League Two side reached the quarter-finals for the first time in 80 years after beating Motherwell in the fourth round in November and Stenhousemuir on Saturday.
Albion will be wary of the League One leaders after they suffered a 4-0 defeat to Ally McCoist's side in the Ramsdens Cup this season.
Jackie McNamara's Dundee United claimed a 2-1 victory over St Mirren in Sunday's noon kick-off to book their place in the last eight where they face an away clash at either League One Stranraer or Inverness, who drew 2-2 on Saturday.
Championship side Raith Rovers have home advantage against St Johnstone as they take on Tommy Wright's Premiership outfit at Stark's Park, having knocked out Hibernian at Easter Road.
All ties will be played on the weekend of March 8-9 with the semi-finals both being played at Ibrox and the final at Celtic Park.
Raith Rovers v St Johnstone, Aberdeen v Dumbarton, Stranraer/Inverness v Dundee United, Rangers v Albion Rovers