A fast start to the game saw Mark Foster get a header on target in Town's first attack, visiting keeper Darren Watts saving comfortably.
With play flowing from end to end and tackles flying in a 20-yard free kick from the lively Paul Goodhand brought another fine save from Watts before the visitors took the lead with a 21st-minute strike from Jed Ainge. Tim Wooding fired a long free kick into the Town area, and when Phil Cassidy played the ball to Ainge the Cogenhoe forward drilled his shot past Town keep Nick Conroy.
The Poachers suffered another blow on the half-hour, leading scorer Gary Bull leaving the pitch to be replaced by Ollie Pinner, but Town kept their heads up, raining in efforts from Foster and sub Pinner.
Boston's efforts were finally rewarded in the 36th minute. The visitors' defence failed to deal with a Goodhand free kick and Matt Kennedy forced the ball over the line after a goalmouth scramble.
Town looked sharper after the break and Watts pulled off a string of saves to deny Foster, Nathan Iley and Kennedy.
In contrast, Conroy was rarely troubled, only an 18-yard Cassidy effort causing him any concern.
Kennedy rounded Watts but then delayed his shot and saw his effort headed off the line by Jack Ashton, but with extra time looming – and in the gathering gloom – the home side again fell victim to a set-piece goal.
Shaun Baker was penalised for hand ball on 83 minutes and Wooding's free kick into the box was met by the unchallenged Nick Ashby, whose deft header found the back of the net to send the visitors into the next round.
TOWN: Conroy, Lambley, Price, Foster, Baker, Iley, Lovas, Goodhand, Smith, Bull (Pinner, 31), Kennedy; subs (not used): Collins, Ogden, Jeffrey, McCallister.
Star man: Paul Goodhand. Another super performance, who ran himself into the ground and did not deserve to end up on the losing side.

TOWN'S second string went down 2-1 away to Lincs League leaders Hykeham Town, despite a spirited performance.
They fell behind early on and then missed the chance to level when Michael Gardner failed to convert from the penalty spot.
The home side notched a second after the break, and although Ben Jacques scored a goal on his debut to reduce the deficit, Town's youngsters could not find the equaliser.