Yet unlike last season's exit to Kent League Erith & Belvedere, Gosport were on top for long periods of the game and should have seen off their Wessex League opponents.
Gosport did suffer an early blow when Aaron Cook limped out of the game with Owen Elias replacing him but Alex Pike's side continued to threaten.
James Taylor saw his penalty claim waived away before Justin Bennett should've done better when well placed.
Jason Silver, making his debut for Boro', then came closest of all as he latched onto Bennett's pass and beat two defenders before scooping his effort over the bar.
After a goalless first period, Gosport got the breakthrough they deserved on the hour when George Way picked out Bennett who beat the advancing Hook to slip the ball into the net.
But Boro' were rocked twice in two minutes which would ultimately lead to their downfall.
Connor Bailey's cross caused havoc in the previously untroubled Gosport defence and Jack Lee poked home the ball into the top corner.
And with Boro' still reeling from the equaliser Steve Hutchings raced onto a ball over the top and lobbed Ryan Pryce to give the Portsmouth side the advantage.
Gosport immediately hit back with Bennett going close to claiming a superb equaliser but the on-form striker saw his shot crack back off the inside of the post.
Then minutes later Taylor had the ball in the net only to see the linesman's flag raised and the goal disallowed.
With Boro' making most of the running, Moneys almost wrapped up the game through Hutchings and Blake late on but ultimately the one goal advantage proved enough and Gosport again miss out on the prize money and prestige of an FA Cup run.
