The Diamonds could not repeat the giant-killing feat which saw them dump Hearts out at the last stage and were outclassed by Craig Levein's men.
Striker David Goodwillie opened the scoring on 32 minutes before Scott Robertson secured the victory with a stunning strike 10 minutes later to send last season's finalists into the last eight.
United were without striker Jon Daly, who scored a hat-trick against Cowdenbeath in the last round, because his wife was in hospital having their first child.
That gave a chance to youngster Goodwillie, who also starred in the last round, and he partnered Spaniard Francisco Sandaza in attack.
Airdrie created a slight opening after three minutes when Paul di Giacomo dispossessed Darren Dods on the edge of the box but the former Kilmarnock striker was not quick enough to beat Lee Wilkie to the ball.
United enjoyed the majority of possession but still had to wait until the 13th minute for their first sight of goal.
Poor defending by the Diamonds allowed Morgaro Gomis to go forward and when his pass to Scott Robertson was intercepted the deflection fell kindly for Sandaza.
The striker, who joined from Valencia in the summer, took an almighty swipe at the ball first time - trying to catch out goalkeeper Stephen Robertson with a volley - but completely missed.
United had another chance on 22 minutes when Gomis was whipped down by Stephen McKenna on the edge of the area but even though Craig Conway did well to loop his free-kick over the wall, Airdrie goalkeeper Stephen Robertson was able to pluck the ball out of the air.
United finally took the lead just after the half-hour, with Goodwillie continuing his impressive form.
A cross from Paul Dixon, making his debut at left-back, found Lee Wilkie and even though Robertson saved his header he was powerless to stop Goodwillie nodding the rebound into the net.
United keeper Lukasz Zaluska sickened the Diamonds on 37 minutes, making an acrobatic save to deny Joe Cardle.
Craig Levein's side then went two ahead just three minutes before the break with Scott Robertson curling a fantastic shot around the goalkeeper from 20 yards.
Robertson nearly added his second of the night with another shot from the edge of the box but this time the goalkeeper was able to save.
Sandaza should have scored for United on 57 minutes when the ball broke to him off the shoulder of Marc Smyth but he snatched at his shot, sending it over.
United were peppering the Diamonds' goal with shots but Gomis, who was outstanding, missed the target from 18 yards.
Levein swapped his strikers with 22 minutes remaining, bringing on on-loan Sunderland forward Roy O'Donovan for Sandaza.
The Irishman tried his luck with an overhead kick almost immediately after coming on but the ball skidded past the post.