The young Gers required a turnaround, trailing 3-2 from the first leg, and they duly delivered an excellent performance as goals from Lewis Stewart, Kyle Glasgow and Max Cameron delivered a 5-3 aggregate triumph.
Speaking to RangersTV after the win, Smith said of his pride: “The message before the game was the team that's going to go through the tie is the team that's going to execute the game plan under pressure.
“And we did that more than what they did.
“They're a better team probably than what they showed, but that was due to how good we were, especially out of possession.
“I spoke to the players after the first leg, and we knew this was going to be difficult.
“We knew we were probably going to have to be perfect, especially out of possession.
“We were going to need to be perfect tactically, which we were for 95 per cent of the game.
"I thought we were really dominant and really good throughout the game.
“It helped that we had a good look at them last week. We felt what the game was like. We felt what the opposition were like.
“We knew they weren't going to change too much as that's been their style in all of their games. But we also knew that we could go up a couple of levels and we did that in the game.”
All three goals were delivered by players currently on co-operation loans, with Stewart plying his trade for Queen’s Park in the Scottish Championship and both Glasgow and Cameron for Alloa Athletic in League One.
Smith believes the result underlines the abundance of talent within the youth ranks at the club.
He added: “As a front three, they probably gave us a bit of flexibility. The three of them can play across the front line in any position.
“We changed that up from the first leg because we've seen a few things that we could probably exploit more so in this leg.
“We put them in the positions they were in, and the three goals were really good goals.
"The fans obviously recognise a few names from the boys now being out on loan and getting that exposure.
“But there are also still good players within the building as well.
"It was a real mix of the loan boys and the boys that are in the building putting on a really good performance together.”
Rangers Under-19s produced a strong performance to overcome Dynamo Kyiv 3-0 in the second leg at Forthbank Stadium, and you can view images from the night below.