In what was hardly a classic the scoreline doesn’t reflect Pompey’s dominance as Marcus Harness’ sixth goal of the season during the first half was enough to seal victory.
The first chance of the afternoon fell to hosts as that man Harness found space on the wing and drilled a teasing ball across the face of goal - but neither John Marquis or Miguel Azeez could make ground to tap into an empty net.
And after a spell of Blues possession, they were given a strict warning that the visitors weren’t in PO4 to make up the numbers when James Ewington beat Raggett for pace before testing Bazunu’s reflexes with a curled effort from inside the box.
Following The Boros’ best chance of the first half, Danny Cowley’s side grew dominant and were patient as the non-league side defended valiantly.
Romeo soon forced Josh Strizovic into a smart save with a powerful effort before turning provider for Harness’ opener moments later.
The Millwall loanee found space down the wing and provided the first real cross of quality when squaring to the 25-year-old – who passed the ball past a despairing Strizovic.
On the stroke of half-time, Pompey were let off the hook again though as Raggett’s undercooked back-pass to his goalkeeper was latched onto by Ewington and as he probed to fire home, Bazunu produced an expertly timed tackle to maintain a clean sheet going into the interval.
Minutes after the re-start Harness picked up from where he left off when he glided into the box and unleashed a low shot that unfortunately kissed the post before Harrow could clear the danger.
With a narrow lead on the hour mark, the match almost changed dramatically when Harrow’s James Mansfield was unmarked at the back-post but he couldn’t make the most of a teasing cross by failing to make true contact with the ball.
The Blues were living dangerously and almost conceded again when they were hit by a counter attack from their own corner as George Moore collected the ball on the edge of the box and fired just wide.
However, Pompey retained a stronghold in the game as Ronan Curtis started to cause the Harrow defence problems.
And with 15 minutes remaining the match opened up slightly as Harrow tried to force a replay, but the hosts held on and booked their place in the second round of the competition, with the draw taking place on Monday.