After early Thatcham dominance, the visitors scored from their first chance on 29 minutes, as Daniel Campion slid in and forced the ball home at the second attempt.
But just five minutes later David Nyarko headed Town level from a Callum Willmoth free-kick by the corner flag.
Thatcham continued to press as Daniel Purple denied Christian Johnson and Harry Hoath, who also flicked a header against the post.
In the second half Thatcham remained on top as Mo Ceesay and Kyle Daniel-Spray fired narrowly over, then Purple came out to block shots from Nathan Jones and Jordan Alves.
Nyarko then saw Purple deflect his shot on to the crossbar as Thatcham continued to be frustrated.
In the closing stages Corinthians launched a rare attack and Ayomide Majekodunmi’s centre was picked up by Campbell who fired in to the far side of goal for an unlikely victory.
TEAM : Fisher, Gwavava, Daniel-Spray, Willmoth, Heholt, Angell, Alves, Jones, Nyarko, Hoath, Johnson
SUBS ON : Ceesay, Jones, Tanner
SUBS UNUSED : Clark, Smith