It was all to play for after a 1-1 draw in Aberdeenshire on Saturday but the Diamonds suffered an early blow with the influential Paul Paton forced off with an injury after just six minutes.
Cove captain Mitch Megginson put the visitors in front on the night and on aggregate with a quarter of an hour played but Calum Gallagher hit back four minutes later to square things once again.
Megginson was then himself replaced after picking up a knock and there was no further scoring in the first period.
Veteran forward Rory McAllister struck what appeared to be the winner in the 92nd minute but substitute Jack McKay levelled again two minutes later to force extra time.
Gallagher then scored his second of the night with ten minutes of extra time remaining to book Airdrie’s place in the final on May 18 while Cove boss Paul Hartley was sent off at the death.
Airdrie face Morton, who overcame a 2-1 first-leg defeat by Montrose to win 4-3 on aggregate.
Gary Oliver and Cameron Salkeld had the Cappielow side 2-0 up after 18 minutes, but Russell McLean pulled one back for the visitors on 34 minutes.
With the aggregate score tied at 3-3 the game entered extra time but with penalties on the horizon, Craig McGuffie scored a screamer in the last minute of extra time to seal a 4-3 victory and a place in the final.