A tenth-minute penalty from Cal Roberts gave the young Magpies the lead at Meadow Lane on Tuesday night, before Danish teenager Elias Sørensen doubled the advantage in the second half.
United had beaten Doncaster Rovers 3-1 in their opening group game and in both matches, the Magpies' pace and purpose on the break was a big factor.
But Dawson told nufc.co.uk: "I thought we looked more solid defensively on Tuesday than we did against Doncaster, probably as a consequence of the work the lads have done over the last few weeks.
"I thought we looked a real threat on the break, and even in possession we were better at managing the game.
"We had a good break in the first ten minutes that led the penalty and we went 1-0 up, but then we had good spells in possession where we managed to get the ball in their half and look after it, make some passes, kill the game off and let them come onto us.
"The first goal obviously made a massive difference because we knew they had to come out, but we managed the game well and the clean sheet was pleasing at the other end.
"It was that siege mentality in and around the box, which we've spoken about recently; just to keep the ball out, people putting their bodies on the line, getting blocks in there, doing the nasty and horrible things.
"And then when we do get it, we do what we're good at. What we have got is real pace so when teams do squeeze up the pitch against us, we know we'll get opportunities with the likes of Cal (Roberts), Tom (Allan), Elias (Sørensen), and Adam (Wilson) coming off the bench.
"We've got a real threat in behind and it was just a case of once we scored the first one, you know you can soak it up a bit and you'll get those chances."
While the Checkatrade Trophy has undoubtedly brought the best out of Newcastle's youngsters this term, Dawson has been pleased with his players in general of late.
They've won all three games this month, with a 5-2 victory over Fulham in Premier League 2 and a 2-1 Premier League Cup success against Notts County's second-string prior to Tuesday's fixture, and Dawson explained: "It was our third game in eight days and we've managed the group in terms of looking after them and wrapped them up when they've needed it.
"The intensity and focus has been good in training and that transferred onto the pitch on Tuesday."
Newcastle now know that they only need to avoid defeat in their final group game at Grimsby next month to qualify for the last 32, but Dawson insisted: "I don't think we'd ever take a back step and sit back and think we just need a point.
"We'll be going with the mindset that we'll try and get our noses in front and then see what happens from there."
Before that, though, United return to Premier League 2 action with a game at Aston Villa on Monday, 22nd October followed by a home match against Middlesbrough on Friday, 26th October.
The venue for the Tyne-Tees derby will be St. James' Park (kick-off 7pm BST) and Dawson said: "You'd like to think at the start of the half term holiday, you might get a decent crowd.
"There was a small number of fans at Notts County (77 die-hards were in the away section) and they made a big effort to get to the game.
"The lads appreciated that and went over at the end to make sure the appreciation was felt."
