Delays to the refurbishment of the Copland Stand had forced Rangers to start the season at Hampden.
Philippe Clement's men have endured a tough time in the opening months of the campaign, falling five points behind Celtic at the top of the Scottish Premiership and crashing out of the Champions League qualifiers.
But there was some home comforts for the Glasgow giants as Cyriel Dessers struck twice to send the holders into the last four.
Dessers opened the scoring by turning in James Tavernier's cross on 18 minutes.
Tavernier's penalty doubled Rangers' lead before Dessers slotted home his seventh goal in nine appearances.
Aberdeen made it 11 wins in 11 games under Jimmy Thelin by ending Spartans' fairytale run to the last eight.
Pape Habib Gueye, Kevin Nisbet, Leighton Clarkson and Ester Sokler were on target for the Dons.
Celtic host Falkirk in the final last eight tie on Sunday.
Motherwell booked their place in the semi-finals with a 2-1 win over Dundee United on Friday.
