The 4-0 scoreline arguably flattered the Light Blues but there can be little question that they earned their place in the draw for the next round of the competition.
Kirkwood said: "I'm delighted with the outcome and with the second half in particular. We had a lot of the ball in the first half without really troubling them too much.
"We got lucky with the first goal and that gave us some breathing space. We changed to be a wee bit more direct in the second half and we're delighted with how the lads adapted to that.
"They got used to the slight change in tactics and we got the three goals to get the victory. It was a tricky tie but the way they went about their business was great.
"The second goal took the sting out of them a wee bit and our lads got stronger and stronger as the game went on."
Kirkwood was also pleased with the contribution of Craig Halkett who grabbed the fourth after being introduced as a forward despite being more regularly used as central defender.
The coach added: "He did well in that role and might have had a hat trick. He allowed us to be that bit more direct and he's one of a few that can play in a couple of different positions.
"Overall though the result is the most pleasing thing. We have a young squad in the competition this year but in each of the rounds the boys have done a good job."