Morgan Feeney's second-half header at Goodison Park sealed a cup double for the Toffees after securing the Premier League 2 Division One title, putting an end to United's respectable run throughout the tournament.
However, Newcastle more than held their own against the home side but were ultimately restricted to a lack of clear-cut goalscoring opportunities and were unable to find a much-needed breakthrough in the tie.
Dawson told nufc.co.uk: "I couldn't be prouder of the lads for the season they've had so far. Tonight was disappointing but I didn't think there was a lot in the game. I thought we controlled the ball really well and they controlled around their goal really well. Neither team really looked like scoring in open play so it was always going to come down to a set play or individual error, which proved to be the case.
"It was a cheap free-kick to concede from which allowed them to get their noses in front. The rules then changed and we were chasing the game a bit. We made a few changes and it looked like we controlled the ball but we didn't seem to have enough in that final third to open up and create a good opportunity to score.
"I thought in terms of the whole game, we were terrific. We had the one scramble on the line early on from a set play, which could have gone in, but overall it summed up the improvement they've made to come and play the division one champions and put in a performance where we didn't look like a team in division two.
"I thought it was more evenly matched. We had good control and built up the pitch. We played out well and flew into that last 30 yards, managing to position ourselves where we could sustain things and keep the ball camped in that area."
Newcastle's run through the competition involved impressive victories over Derby County, Southampton and Leeds United and, after falling at the final hurdle, Dawson believes the difficult defeat can be used in the young Magpies' favour.
United's second-string travel to Southampton for their final game of the season, competing at St. Marys' Stadium in the Premier League 2 division two play-off final as the youngsters bid to end their positive campaign on a high.
Dawson added: "The message was that you'll be massively disappointed, as everybody is at this moment in time, but you've got to forget about it now. We've got to focus on the next game which becomes even important.
"I'd like to think the defeat can spur the players on. In terms of the next game, you'd like to think that it will be another great memory that they don't want to experience again. We're looking ahead to five or ten years' time when, hopefully, the first team are in a cup final and some of our lads are playing who've experienced that tonight and they know what the pitfalls are and what it feels like to get beat, so hopefully they can learn from that."
