The Blues’ 2-1 win at Rovers on Saturday booked a place in the last 16 against Tamworth or Corby, on the Northolme, on 12th January.
Stamp’s deft opener on 15 minutes gave Trinity a merited lead, and the experienced striker bundled home a second with 18 minutes left to stun the Conference promotion hopefuls.
Chris Stokes’ goal on 79 minutes made for a barnstorming finish, but Housham’s in-form side sealed their sixth win on the trot and a shock victory.
“That’s six on the bounce now and one defeat in 12, so the run goes on and the dream goes on,” said Housham.
“People kept telling me during the week what a good side Forest Green are, but we’re a good side and we came with no fear. I’m as proud as punch.”
The underdogs certainly looked far from overawed during the opening stages, Luke Waterfall getting on the end of a ninth minute free-kick and sending a header wide of Sam Russell’s far post.
Six minutes later they were ahead, Stamp reacting quicker than Aarran Racine to pounce on a ball over the top before expertly lifting the ball over the onrushing Russell and into the net.
Russell’s opposite number Jan Budtz was tested for the first time by a Steve Brogan free-kick, which was curled around the wall and forced the keeper to plunge to his right and parry.
Cue a spell of pressure from Dave Hockaday’s side, as captain Jamie Collins soared to meet Brogan’s corner with a firm header and was denied by Dominic Roma’s clearance off the line.
Trinity survived to regain a foothold in the contest, and they could even have doubled their advantage three minutes before half-time when Josh Wilde’s superb cross found Stamp and he nodded wide from ten yards out.
Moments later Wilde’s cross-shot had to be helped around the post by Russell, and from the resulting corner Craig Nelthorpe blazed a decent opportunity over.
Trinity started the second half on the front foot, Nelthorpe embarking on a mazy run and unleashing a dipping shot that landed on the roof of Russell’s net.
Belief was growing in the away ranks, and it surged through the roof in the 72nd minute when Daniel Hone rose highest to nod down a corner and Stamp was on hand to ram home his second goal of the afternoon.
The lead was halved seven minutes later, Matty Taylor pipping Budtz to the ball and digging out a deep cross for Stokes net at the far post.
Rovers Substitute Yan Klukowski thought he had equalised, only for Wilde to nod off the goal-line, and Collins was wayward with a header under pressure.
Carnage from two corners at the death was somehow negotiated by Housham’s men, who march into the last 16.
Forest Green: Russell; Turley, Racine, Oshodi, Stokes; Collins, Bangura, Brogan (Klukowski 66); Vieira (Norwood 46), Styche (Taylor 46), Koroma; Subs not used: Hodgkiss, Jarvis.
Gainsborough: Budtz; Roma, Waterfall, Hone, Wilde; Hawkridge (Yates 65), D’Laryea, Leary, Nelthorpe; Knowles, Stamp (Barraclough 81); Subs not used: Young, Wilson, Connor.
Goals: Stokes 79; Stamp 15, 72.
Booked: Styche; Wilde, Hone, Leary.
Star man: Daniel Hone (Gainsborough).
Referee: Justin Amey (Dorset)