Philippe Clement's side were made to pay for spurning numerous first-half opportunities and now look set to end another season without a trophy.
Queen's Park goalkeeper Ferrie made a couple of excellent stops to deny both Mohamed Diomande and winger Vaclav Cerny, before Zach Mauchin headed Ianis Hagi's goalbound effort off the line just before the interval.
The visitors grew into the match in the second period and came within inches of an opening goal through Kyle Hirst before Drozd fired them in front.
Rangers pushed for a leveller and were thrown a lifeline deep into injury time as Cyriel Dessers won a late penalty after being pushed in the box.
But captain Tavernier was denied from the spot by the heroic Ferrie, who dived full-length to his left to tip the ball around a post and earn Queen's Park a first appearance in the last eight since 1983.
Clement's dreadful Rangers lacked creativity and imagination in the final third and were loudly booed off the pitch at full-time by their supporters.
Queen's Park manager Callum Davidson now returns to the quarter-finals for the first time since knocking out Rangers at Ibrox on his way to cup glory with St Johnstone in 2021.