The Deans were much the better side over the course of the 90 minutes as it finished 1-1, only for the visitors to take victory courtesy of a 4-1 shoot-out triumph in the second round qualifying tie.
Chad Field had fired Horndean ahead 10 minutes after the restart before the hosts saw Jack Lee dismissed for a foul in the area on the hour and Silvano Obeng dispatched the resulting penalty.
But Birmingham felt - even though his side had a man disadvantage for the final half-hour - it was his troops who were the better team throughout.
And the Horndean boss reckons his men did enough to run out winners in normal time before going down on penalties.
Birmingham said: ‘I think it was a little bit unfortunate (penalties defeat) but I’m not going to moan, the lads gave me everything, I thought they were absolutely outstanding.
‘Even when we went down to 10-men, in my opinion, we were the team who were looking for the goal and we always looked a threat. I thought some of the football we played with 10-men was absolutely outstanding. We just got stronger and stronger.
‘It happens in football, it’s a cruel game, you can turn around and say did we do enough to win the game? I think possibly we did - let’s not take nothing away from Brockenhurst - they had a game plan, they tried to frustrate - at times I didn’t think it worked - we really nullified them all over the park.
‘We gave away a silly penalty - it was a silly penalty - Jack Lee put his hands up and said it was a silly penalty. I’ve got no dramas - it was a stonewall penalty.’
With penalties looming, Birmingham opted to take goalkeeper Cameron Scoot off and replace him with former Pompey stopper Leon Pitman in the final minutes of normal time.
The Horndean boss explained he felt it would provide his side with a better opportunity of claiming victory in the shoot-out.
He added: ‘Jocko (Cameron Scott) has been outstanding but I think Jocko would be the first person to admit Leon is a bit more springier than him.
‘I didn’t expect anything different to Jocko’s reaction - I wouldn’t want to be taken off at the end of the game.
‘He’s a good, honest kid, I did it for the right reasons for the football club, it was nothing personal.’