The Light Blues won 2-1 thanks to Andy Murdoch’s screaming 18-yard strike after home striker Callum Gallagher’s opener was cancelled out by Liam Caddis.
That was despite going into the game having not yet played a competitive game, whereas Saints already have three league matches behind them.
In addition, at least eight of the away side’s starting line-up were at least 18 while the hosts used nine 16 and 17-year-olds during the 90 minutes.
Kirkwood was thrilled with the final outcome and paid tribute to replacements Dylan Dykes and Jordan Wilson for helping set up the winner seconds after the came on.
The coach said: “We’re very happy to progress, especially when you consider some of our lads were giving away two years of an age gap.
“They only had one outfield player born in 1995 in their starting team and we had a lot more than that so we’re delighted with how the young lads performed.
“Some of our lads are first year full-timers and their bodies are still adapting to that. The day in, day out nature of the job can really take its toll on their strength at first.
“It is a hard ask of them but they’ve got to show they’re able to take it and so far they are doing that.
“We scored a terrific opener and then reacted really well to St Johnstone equalising. It was fantastic and the two subs we sent on were part of build-up to our winner.
“It was a hard-fought game. It was a tough tie for both teams because in the first round of the cup the bigger teams don’t tend to meet each other.
“Instead they come up against the smaller sides and they ease their way through the first couple of rounds quite comfortably so we’re pleased we’re in the hat.”
Murdoch’s strike was outstanding and it gave away keeper Craig Reid no chance as it soared into the net 10 minutes from the final whistle.
Kirkwood added: “It was a really sweet strike. It maybe took a wee deflection on its way but he hit it ever so well.
“The young kids want to play at Ibrox so there were a few disappointed faces when the team was read out so it was good Jordan and Dylan went on and played a part there.
“Them and young Danny Stoney who also came on helped us into the next round and sometimes one man’s poison is another man’s meat.”