Tom Allan continued a memorable week as the Newcastle-born forward, who made his first-team debut in United's Emirates FA Cup win against Rochdale, broke the deadlock in the Group C clash with a first-half penalty, netting his 12th goal of the season in the process.
However, Newcastle's youngsters, who reached the Premier League Cup final last term, were made to rue their second half misfortunes in front of goal as Danny Lupano's close-range header for the Tigers saw the shares spoiled at Whitley Park.
"We're disappointed after creating a lot of chances, especially in the second half," Dawson told nufc.co.uk. "We probably should have had the game dead and buried. From that point on, we got caught up emotionally where you are frustrated that you haven't scored so then you're almost trying to fix your mistakes by doing something different.
"We had too many trying to do their own thing and not fulfilling the role as part of the team and doing what they know works in what they do week in week out. That's the disappointing thing and it leads to them getting back into the game because we're not set up behind the ball like we should be.
"At one nil and then 1-1 each, if you don't take your chances, you always run the risk of them maybe getting a set play or getting up the pitch on a breakaway and causing us problems, which they did.
"I said to them before the game, we're sitting in second of the group at the minute but you're going to need to pull your finger out. We had a great performance against Fulham and then a 60-minute performance away at Manchester United after the first half hour.
"We haven't really had a performance there today and we've come away with a point but they are in that position now in the group where they need three points just to make sure otherwise the season can very quickly peter out into nothing."
Following Allan's professional debut at St. James' Park on Tuesday night, Dawson was pleased for the 20-year-old in featuring for his boyhood club at senior level.
Allan became the third Academy product to feature in a competitive fixture for United's first team in the past 12 months after the breakthrough of midfield duo Sean and Matty Longstaff.
"Tom's been terrific," Dawson added. "We brought him off (after 65 minutes) as he looked tired both physically and mentally. I was speaking to some of the staff and saying when you make your debut, on a night time game when you have had all day to think about it, he's almost emotionally drained from running that game through in his head two or three times.
"As much as he may be tired, it's been a great week for him along with everyone involved at the Academy to get another one on the pitch.
"A lot of it is down to them, making sure they are doing the right things everyday in training and when they do get the chance, they grab that opportunity to stick around just like Sean did last season and like Matty managed to do in the summer."