The tie had finished 1-1 after extra-time, with Connah’s winning through to face Ross County after John Danby saved efforts from Craig Thomson and Ciaran Diver.
Both County and Connah’s will need to travel over 200 miles to find neutral territory for the 23 or 24 March showdown.
Afterwards, the Welsh club’s brusque Scottish manager Andy Morrison was unconcerned with winning friends in his homeland, recalling the feisty win over Queen’s Park earlier in the tournament.
Morrison said: “Ross County are good, but we will look to stifle them. If it was a two-legged final, I would be worried, however it is a one-off and our boys can achieve anything.
“We stood up to the men of Falkirk and then beat the monsters of Coleraine. They were a big physical side who had gone 35 games without defeat.
“We then beat a big bunch of girls at Queen’s Park. I was delighted with that as they showed us no respect.
“Funnily enough they lost a lot of games after playing us as they were still crying like babies.”
City opened in stunning fashion, with Josh Walker firing in a rebound with less than two minutes played, but Michael Wilde netted for Connah’s from six yards after getting on the end of a Jack Phillips throw.
Phillips came close to a winner but was denied by visiting keeper Callum Antell.
City boss James McDonaugh said: “It was a cruel way to lose but I have nothing but pride for my players. We were huge underdogs and took it all the way to penalties.”