The young Magpies fell behind by two goals on the hour mark, as Sam Osborne's first-half strike followed by a ferocious volley from Elliot Ward gave County a clear advantage.
However, Mo Sangare pulled one back for United before substitute Kelland Watts netted a late equaliser with a looping header from a cross by Rosaire Longelo, who made an impressive impact from the bench with both assists.
Although Dawson was disappointed not to secure maximum points, the under-23s coach reflected positively on the team's fighting spirit as Newcastle battled for a point to go top of Group G on eight points.
"We had spells of really good play in possession and controlling the game with or without the ball," Dawson told nufc.co.uk. "Their first goal both come from us trying too hard to score instead of maintaining control in the opposition's half.
"We were not set behind the ball and made poor decisions. At half-time, we then had to try and rectify a few things but the one thing the lads have shown is their character and resilience to get back into the game. It shouldn't have to come to that and the group know that it wasn't great.
"We've managed to scrap out a point but going forward we've said there is a few games now where we haven't dealt with the ugly side of the game in dealing with second balls, not conceding cheap set plays, flick-ons and anticipating where it will drop.
"As you go up the levels, they can be so important and you have to earn the right to play the good football we want to and, unfortunately, we didn't do that enough throughout the 90 minutes."
With Leeds United and Southampton left to face in the group stage, Newcastle know a win in either fixture will send the Magpies into the knockout stages.
And Dawson was keen to highlight the determined attitude shown by his squad in recent weeks, with the second string managing to come from behind to claim a positive result on four occasions in their last six matches.
"The lads always have that winning mentality," Dawson added. "I think people get confused between winning games and if you got beat that you've haven't shown it. It's always there and something we work on in training, which is always competitive with people wanting to win all the time.
"I'd have been disappointed if we'd thrown in the towel after going two-nil down as it would have displayed a lack of winning mentality. We need to focus more on the quality of decision-making and performance both as individuals and as a team.
"Regardless of today's result, we always knew that the games against Southampton and Leeds would be important. We knew we couldn't qualify from the group today so we knew we had to pick up points from those games and that's the same situation we're in now.
"We look ahead to a tough league game against Wolves and then the Senior Cup tie (against Blyth Spartans) so we will focus on those two cup games when we get to them."
