United reached the last eight of the competition in 2017/18, as well as progressing to the semi-finals of the Premier League International Cup and lifting the Techflow Marine Northumberland Senior Cup and Hong Kong Football Club Citi Soccer Sevens.
And, having already kicked off the Checkatrade Trophy with a 3-1 win at Doncaster Rovers last month ahead of next week's trip Meadow Lane to face Notts County's first team, Dawson was pleased to see off the other Magpies' second-string at Whitley Park.
"We got to the latter stages in all the competitions last year so it's good to get off to a good start with three points," Dawson told nufc.co.uk.
"Our performance level probably wasn't up to where we expect but it was our second game in four days and we've got another one coming up in the Checkatrade Trophy, another cup competition, so I would imagine one or two of the boys maybe had a little bit of an eye on Tuesday.
"But the professional element of tonight was we found a way when we weren't playing well, weren't in control of everything we wanted to control; we found a way to get off to a good start.
"It was a lot more direct and a lot more physical compared to Monday, against Fulham, when they tried to pick holes and find a way through on the deck. Notts County still had that - they had that element of making a few passes, playing between the lines and causing a few problems that way - but they also had the option to go more direct and put our back four under pressure.
"That led to their goal, where we switched off for a split second and allowed them to get in, but hopefully they'll learn from that. I'd imagine Tuesday will be a very similar shape and style of play from their first team and we'll have to be big enough and strong enough to stand up to anything that does get thrown in, but then find a way to impose our game."
Morocco international Achraf Lazaar set up the opening goal for Elias Sørensen, having claimed two assists in his first second-string appearance of the season against Fulham on Monday.
"When we do get the ball into those wide areas, you can see the quality he's got in terms of his delivery," explained Dawson. "He's got that knack of being in the right place at the right time and it puts people under pressure, the deliveries he puts in.
"We're just trying to get used to the quality that he brings. A lot of the time you're just thinking 'keep it here' and then he puts it on a sixpence for somebody so it's been good in terms of having him around the group. He's an enthusiastic and bubbly character around the place and he gets everybody going in the morning, big smile, but he's not afraid to take it on the chin when the lads have a bit of a go at him, so it's been good."
It was Danish teenager Sørensen's seventh goal of the season - he hit a hat-trick against Fulham - and Dawson said: "His workrate is improving game by game and his understanding out of possession is improving game by game.
"We need a little bit more from him in terms of looking after the ball, getting his back in and sometimes just winning fouls for us to get us up the pitch. But when we do get the ball in and around the box he's got that knack at the moment of finding the back of the net. Hopefully he does the same again on Tuesday."
Cal Roberts scored the crucial second before Josh Bingham pulled one back for the visitors. Roberts was involved with Rafa Benítez's first team in pre-season and has been nominated for September's Premier League 2 player of the month award, and his latest goal was a stunning strike after a driving run.
"It was a great finish," said Dawson. "We've been trying to manage Cal, just with a little bit of an injury, but it was important that he got some time on the pitch tonight. It was a slightly different position, just to help get him through that, but it was a terrific run and strike for the goal.
"He needs to take more responsibility on his shoulders now. That's the level he has to get to every week, every game. It can't just be once or twice a month - it has to be three games in eight days. That's the level he has to get to.
"But he's showing a little bit more maturity, being one of the more senior lads in the group; he's having to become more outspoken, the younger boys around him are demanding more from him because they know what he can do. He almost can't look up to anybody else now and demand off them - he's one of those 'senior' players taking that responsibility on. Like I say, he just needs to do it a little bit more often."