Diamonds fought back from 1-0 down in normal time and 3-1 down in the second period of extra-time with Michael Rankine scoring a dramatic equaliser with almost the last touch of the game.
However, it was the Shots who held their nerve as Andy Gooding and
skipper Chris Hope both missed from 12 yards for the visitors.
Despite the defeat, Hill was delighted with the character his side showed.
Speaking after the match he told Setanta, "Whenever you lose a cup final it's going to be disappointing but I am proud of the players.
"We had a lot of young lads out there and I felt we showed a lot of character.
"We have been beaten by the side who will be league champions and league and cup double winners.
"They are a good side, but from a manager's point of view I think
anyone who would have watched the game wouldn't have believed that
Aldershot are 41 points in front of us."
Hill, meanwhile, was left to reflect on his side's missed chances as Rankine headed wide when left completely unmarked in the final minute of stoppage-time in the first 90 minutes.
But he paid tribute to Aldershot who are, in their words, now likely to go on and claim their place in the Football League.
"I am proud but I honestly don't think we should have gone to extra-time," Hill added. "We had enough chances to win the game in normal time.
"But good luck to their manager Gary Waddock. He has conducted
himself well all season and his team are a real credit to him.
"They are good side and one good thing that has come out of this season is that we will not have to play them again in the next one."
Source: http://www.northamptonchron.co.uk/