It was a tough test for the League One outfit in difficult conditions and a 3G playing surface which they found hard to adjust to.
However, stunning strikes from Brandon Haunstrup and Ronan Curtis earned passage into the second round, where Pompey will now play Altrincham.
And Jackett was delighted with how they handed their Wetherby Road challenge.
He said: ‘It was always going to be a difficult tie and Harrogate are going well so far this season, genuinely a good team with a way about them that works well.
‘The ball into the centre-forward and then the runners off that caused us problems in the first-half.
‘We got to grips with it slightly in the second and there was a minimal amount of chances, but early on we didn’t quite get a hold of that. The ball coming down the slope with the elements behind them gave them momentum.
‘They got a goal but I am pleased we scored two excellent goals and, as the game wore on, we showed more and more resilience about us to close it off.
‘There's good character in the group. Although there are certain things we could have done better last night, it was always going to be a difficult tie.
‘My thanks to the supporters for coming this far, being patient while the game was delayed and, on behalf of all of the players and staff, we really do appreciate their efforts, they got right behind the boys.’