Goals from United skipper Owen Bailey and Matty Longstaff propelled the young Magpies into a two-goal lead at Whitley Park before Derby replied through second-half strikes by Calum McDonald and Josh Shonibare.
However, in the opening stages of extra time, Mo Sangare sealed Newcastle's place in the last four as the Liberian reacted quickest to Bailey's saved penalty to slot the rebound home.
The victory sees Dawson's side advance into the semi-finals of the competition for the first time, after losing at the quarter-final stage in consecutive seasons but the under-23s coach was delighted to progress at the third time of asking.
"I'm really pleased," Dawson told nufc.co.uk. "It's a cup competition that we want to try and go on to win. The result is more important than the performance and there were some good examples of staying in the game and staying together.
"In the first half, we showed a bit of quality which led to some chances and we should have put the game to bed at the start of the the second half. We had some great opportunities in the first ten minutes and we then allowed Derby to play their game and drop off.
"There was then good togetherness and resilience again once we went into extra time to get our noses back in front against a good team. As a group of lads, they do everything together. They like spending time with each other which creates that close-knit relationship that they've got which you see on the pitch."
Dawson's side enjoyed a spirited run to the last 16 of the Checkatrade Trophy and reached the semi-finals of the Northumberland Senior Cup, losing to Sunderland and Morpeth Town respectively.
After recent near-misses, Newcastle's youngsters dug deep after 120 gruelling minutes against the Rams, who sit in third place of Premier League 2's top tier - one division above the young Magpies.
Dawson added: "I was pleased with the response after conceding two goals. We mentioned the Wolves and Southampton games where we were faced with similar situations and conditions where the wind was blowing from one end of the pitch to another.
"We wanted them to keep doing what they did well in the first half but sometimes the opposition have other ideas. We did enough to stay in the game and stick together to grind out a strong result.
"I think there's a good chance of lifting the cup. It's a cup competition and a one-off game so it doesn't which league you are in - it's two teams against each other for 120 minutes.
"We had the disappointment of the Checkatrade Trophy result and the Senior Cup result, so it was a case of saying fo you want something to play for before the end of the season. That put the onus and the pressure back onto the players and they have responded really well.
"It gives us an extra game to play. It will take us up to 39 games for the season which will be really good. We wanted more games for this group to make it closer to first-team football. Newcastle's first team play 38 games plus their cup matches so we have to try to bridge that gap from under-23 football so staying in the cup is important."
