Jamie Sterry - who returned from a loan spell at Crewe Alexandra this week - opened the scoring at Woking's Kingfield Stadium with a tremendous long range strike before the Royals quickly levelled through Josh Barrett's free kick.
However, Tom Allan restored the young Magpies' lead in the last 16 clash with a well-taken finish before United's youngsters held firm in the second half to seal their passage into the quarter-finals for a successive year.
Before the cup clash, Dawson revealed an important message to the group which the coach believes the squad took fully on board.
He told nufc.co.uk: "I thought we grew into the game well, especially once we were given some encouragement. We created a lot of good opportunities by pressing high and luckily those missed opportunities didn't matter. We managed the game well after going 2-1 ahead on a difficult pitch and in difficult conditions.
"We reminded our players that in order to reach the top level you have to take risks, not to be afraid of mistakes and be willing to take the opportunity to progress in the competition. They did that well from the first whistle and consequently showed a much improved performance from what we have produced recently."
Newcastle's second string achieved an impressive Premier League 2 victory against Reading, beating the Royals 2-0 in February after withstanding strong attacking pressure from the home side.
Once again, Dawson was pleased with the defensive work shown against the Berkshire club and, with Sterry getting his name on the scoresheet, admitted his delight for the right-back's opener after a difficult loan spell at Crewe.
Dawson added: "I'm pleased for Jamie as he has had a tough time recently. We've made the decision to bring him back to the club and it was nice to see him score a well worked goal.
"The goal we conceded was a little disappointing as it came from a debatable free kick. However, the players didn't let that affect them and probably gave them added impetus for the remainder of the first half period.
"I thought defensively we looked strong today. The players at the back controlled the spaces in front and behind the line well. The front five worked together to press up the pitch and, at times, sustain a high press to force a number of errors and provide us some great opportunities to kill the game off.
"Our immediate focus now is recovering well and preparing for an important Premier League 2 fixture with Wolves. We'll obviously watch the draw for the next round and hope that when that game comes around that we can progress into the semi-final."
