Boro' hadn't completed a fixture since their 3-0 loss to Havant & Waterlooville on Christmas day and looked rusty in the early stages.
Hungerford, who themselves had been hit by the terrible winter weather, started the game the brightest, but failed to test Nathan Ashmore in the Gosport goal.
Borough's Danny Smith almost broke the deadlock on the ten minute mark when a sweetly struck volley from the edge of the area, was superbly saved by Paul Strudley in the home side's goal.
Mid-way through the first half, Hungerford's Adam Draycott saw his effort from six-yards out fantastically saved by Ashmore in what was arguably the chance-of-the-half.
Minutes later Ashmore's goal was under threat again.
A cross from the impressive Alan O'Brien was met at the back post by striker Stefan Brown, whose header from two-yards out went wide, much to the relief of the travelling Boro' fans.
On the stroke of half-time, Tom Dunford rattled the Hungerford crossbar.
A deep cross from Sammy Igoe was met by the head of Dan Wooden whose header was saved by Strudley, only for Dunford to rifle the rebound onto the post and out.
The second-half began in the same fashion as the first, with Hungerford doing the majority of the pressing.
On the hour mark, midfielder Dean Stow dragged his effort past Ashmore's post following a one-on-one.
This appeared to be a wake up call for Boro, who four minutes later took the lead.
A free-kick from Igoe mid-way into the Hungerford half was met on the volley by Jamie Brown, who guided the ball into the top left corner of the goal, leaving the 'keeper helpless.
Ten minutes from the end, Hungerford were awarded a dubious penalty following a coming together in Borough's area.
Captain Matt Day stepped up, but was denied thanks to a wonderful diving save from Ashmore.
The remainder of the game was dominated by Hungerford who continued to push for the equalizer until the final minute, but Gosport hung on to earn a deserved win and a place in the Trophy's final eight.
The draw for the quarter-final takes place on Monday (13th January) at midday.