The Portsmouth Football Association delivered the news after meeting on Thursday night.
Borough were found guilty of not fulfilling their Portsmouth Senior Cup tie against the Hawks during the week beginning Monday, October 24.
The tie was scheduled for Tuesday, October 25.
But Hawks secretary Trevor Brock received an email from Gosport on the preceding Saturday stating they were unable to fulfil the fixture that week because of injuries and other commitments.
This was contrary to Portsmouth Senior Cup ruling 13e, which states a game can only be postponed if a club has a match in another competition that takes precedence.
That wasn’t the case, although Borough’s development squad did beat Winchester City 6-0 in a Wyvern Combination League midweek division fixture on the Wednesday night.
Borough boss Alex Pike argued he didn’t know anything about the game and no date had been agreed between the two clubs.
But the Portsmouth FA found in favour of the Hawks, who will now progress to the quarter-finals.
Borough were not represented at the meeting, choosing to outline their case via email.
Hawks chairman Derek Pope was pleased with the outcome.
‘If you enter a competition you have to abide by it’s rules,’ he said.
‘It is an odd situation because they played a game the following evening.
‘Gosport didn’t even turn up at the meeting to fight their cause.
‘What sort of representation of the club is that?
‘Nobody wants to see anyone thrown out of a competition – it should be decided on the football pitch.
‘Unfortunately those are the rules.
‘It is a great competition because it allows us to give the youngsters a chance to play alongside some of the senior squad players.
‘And players coming back from injury also have a chance to get some minutes.
‘We like taking part in it but – because of our other commitments – need to play the games on the set dates where possible.’