Ryan Dow scored his first goal for the club on 11 minutes after Paul McKane had failed to hold a Stuart Armstrong shot.
Ryan Donnelly missed Airdrie's best chance when he failed to find the net after rounding Dusan Pernis midway through the first half and John Daly finished the tie when he slotted home Paul Dixon's cross 15 minutes from time.
The visitors had been forced into a change from the original starting line-up before kick-off with the injured Scott Robertson being replaced by Armstrong and the Scotland Under-21 international was instrumental in the opening goal.
Armstrong broke past two Airdrie challenges in midfield and when his low shot was spilled by McKane, Dow was on hand to round the goalkeeper and slot into the empty net.
McKane was having trouble with shots from distance and after he spilled another long-range effort from Dixon, Lauri Dalla Valle had the chance to make it two but McKane redeemed himself by blocking the Finn's low shot.
Dalla Valle's movement was causing the home defence problems and a simple headed clearance from Gavin Gunning found him through on goal again but after showing good strength his shot lacked power and was easily held this time by McKane.
Airdrie's best chance of the first half came on 22 minutes when Donnelly was played in behind the United defence but after rounding Pernis the angle was too acute and he fired into the side-netting.
United were stroking the ball around with ease in midfield but were not creating the chances they had early in the game and Airdrie looked the likelier side to score as the first period drew to a close. John Boyle came the closest, but after Garry Kenneth's sliced clearance had fallen perfectly for the striker he fired across Pernis and wide of the far post.
United controlled the second half despite never clicking into top gear. John Rankin shot just wide on 58 minutes before McKane made amends for his error at the goal by making a brilliant save to turn away Dalla Valle's volley.
Visiting boss Peter Houston brought on captain Daly for Armstrong in an attempt to put the tie to bed and he nearly made an immediate impact but his header floated over.
Rankin played in another substitute Gary Mackay-Steven but he was denied a goal against his former club by another block from McKane and Dalla Valle ballooned the rebound over.
Daly then did seal the tie with a cool sidefooted finish at the back post from an excellent cross by Dixon with 15 minutes remaining.
Left-back Chris Malone had one final chance for Airdrie but he shot over deep into stoppage time.