The Dons face a midweek trip to Ibrox after Joe Worrall’s second-half goal cancelled out Sam Cosgrove’s first-half penalty at Pittodrie on Sunday.
The winners of the replay will face the treble-chasing Celtic, who defeated Hibs 2-0 in Neil Lennon’s return to the Capital following his departure from the Easter Road side in January.
Hearts will face Championship side Inverness in the last four should they overcome Partick Thistle in their quarter-final replay.
The Jambos were held by the Jags in Monday’s clash at Firhill after Christophe Berra’s opener was cancelled out by Christie Elliott.
The Caley Jags scored a last-minute winner at Tannadice to knock out fellow Championship side Dundee United on Sunday.
The semi-finals will be played at Hampden Park on the weekend of April 13/14.