United's youngsters were held to a 1-1 draw at St. James' Park on Tuesday night after Elias Sørensen's 18th goal of the season was cancelled out by Scott Wilson's delicate chip five minutes from time, sending the game straight to penalties.
However, Nathan Harker's fantastic save from Ben Stephens, along with five well-taken penalties, ensured the Magpies' victory in Sol Campbell's managerial debut for the League Two side and saw United reach the last 16 of the competition.
After a conceding such a late goal, Dawson felt his side managed to dig deep in difficult circumstances to become the first side to defeat the Silkmen on spot kicks in their 144-year history.
"We did it the tough way and had to show effort, resilience and togetherness," Dawson told nufc.co.uk. "We played against a Macclesfield side who were well organised and had an added incentive of playing for a new manager at St. James' Park so, considering there wasn't much quality in the game, I am happy that we stuck together and found a way to win."
After failing to hold onto the lead, Dawson revealed that a great deal of work had gone into spot kicks since the start of the season with both the penalty-takers and goalkeepers in his squad.
He added: "We were disappointed to concede late on with the ball going through the middle of the pitch after moving into a defence of five. Once the whistle had gone for full time, there's nothing you can do about it and we told them to concentrate on the job at hand.
"We have practiced penalties since pre-season, recording every single one, so the squad all have their own bit of paper and know what their best penalty is.
"It was a case of making sure they were focusing on the goal and getting a good connection from the strike.
"Then, we had to rely on Nathan (Harker) to pull one out of the bag, which he managed to do. It was a great save after waiting for the striker to make his decision before diving.
"Not only have we worked on our own penalties but we've also worked on our opponents'. The devil is in the detail in making sure that we left no stone unturned."