The match appeared to be heading for a 1-1 draw as it entered extra-time, after Joe Morrell had cancelled out Rafa Ramos’ opener for the hosts.
However, in stoppage time, Harry Jewitt-White headed home from close range and then Destiny Ojo raced clear to make it 3-1 moments later in front of a 2,250 Privett Park crowd.
John Mousinho used 21 players, with only Will Norris completing the full 90 minutes, as he made nine substitutions of 64 minutes.
The late finale flattered Pompey, but here are our player ratings...
13. Jordan Archer - 5
(Replaced Will Norris on 46 mins) Largely had nothing to do, but handed Gosport a lifeline on 86 minutes when he failed to hold Wooden's free-kick and Yearwood followed up to make it 2-1. Definitely a goalkeeping mistake. Photo: Habibur Rahman
14. Terry Devlin - 7
(Replaced Jordan Williams on 46 mins) Asked to play at right-back and produced a number of attacking bursts down the right. Always has plenty of energy and loves being in the thick of the action. Photo: Jason Brown
15. Conor Shaughnessy - 7
(Replaced Jamal Baptiste on 46 mins) Operated on the right of the centre of defence and looked as composed and classy as ever. Always uses the ball so well and keen to join in attacks. Photo: Jason Brown
16. Tom McIntyre - 7
(Replaced Ryley Towler on 46 mins) A first outing since breaking his ankle in January, McIntyre operated as left-sided centre-half and looked very comfortable. Clearly can cope with the ball at his feet and there were a couple of excellent passes out from the back. Photo: Portsmouth FC