Two goals from Jon Daly (48 and 51) and a late strike from David Templeton (80) ensured Ally McCoist's side will be in Tuesday's draw on an emotional evening in Glasgow.
The Rangers fans began the night with a noisy second minute tribute to former player Fernando Ricksen, who was this week diagnosed with Motor Neurone Disease.
However, it proved to be the biggest cheer of the first half as the home side struggled to turn their near-total domination into chances.
Nicky Law and Bilal Moshni went closest for Gers but it was goalless at the break with Airdrieonians growing in confidence.
That all changed three minutes into the second half when Rangers took the lead, Daly curling in a superb effort after Arnold Peralta's ball into the box had only been half cleared.
And it was 2-0 three minutes later, Daly heading home Richard Foster's cross to put daylight between the sides.
An indirect free-kick on 64, bizarrely awarded after an Airdrieonians defender smothered the ball with his body failed to deliver a third goal - but the 22,533 did not have long to wait.
Templeton obliged 10 minutes from time, stroking home Law's cut-back to end the game as a meaningful contest and send Rangers through.