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...
17. Jack Sparkes - 7
(Replaced Connor Ogilvie on 46 mins) Produced a wonderful goal on the stroke of the full-time to make it 3-1. Crashed a 25-yard left-footed shot into the top corner of the net with the last kick on the game. In addition, always willing to attack down the left and looked good. Photo: Jason Brown
18. Tom Lowery - 7
(Replaced Marlon Pack on 46 mins) Provided a late scare when he conceded the free-kick which enabled Gosport to pull one back to make it 2-1 in the 86th minute. Still, he uses the ball so well and is such an impressive player. There was one injury scare, but he was fine. Photo: Jason Brown/ProSportsImages
19. Owen Moxon - 7
(Replaced Ben Stevenson on 46 mins) Buzzed around and at the centre of plenty of good Pompey moments. There was also a decent shot from the energetic midfielder who has a big season ahead. Photo: Jason Brown
20. Paddy Lane - 7
(Replaced Anthony Scully on 46 mins) Will be disappointed not to get on the scoresheet, but so lively. Saw a one-on-one saved and another chance go close as he buzzed around causing problems. Very bright. Photo: Jason Brown