Despite going behind early in the second period, goals from Leon Broadhurst, Luke Benbow and substitute Josh Green were enough for the Glassboys to progress.
The hosts went close early on from a corner when Taylor Parr’s header went narrowly wide.
Then his central defensive partner Liam McDevitt saw a shot well blocked by Stour skipper Paul McCone before he fired his second effort wide.
Kyle Perry had the first Stour chance from a Luke Shearer cross but the striker’s header was pushed round the post by St Neots keeper Mike Emery.
The crowd thought St Neots had taken the lead 13 minutes in but Johnny Herd’s free-kick had in fact gone into the side netting.
Perry tested Emery again seven minutes before half time, having a 20-yard strike well saved following good work from McCone and Darryl Westlake.
While the first half may have been slightly lacklustre, the second half exploded into action straight away.
Firstly, St Neots took the lead a minute in as Tom Meechan turned the ball inside to Dion Sembie-Ferris who fired home with a low drive.
But Stour replied almost instantly. Tom Tonks’ left-wing corner was flicked on inadvertently by home defender Johnny Herd at the near post and fell for Broadhurst to direct home a header in the 48th minute.
The turnaround was complete just four minutes later. Aris Christophorou was bundled over in the penalty area, the referee pointed to the spot much to the frustration of the home side and Benbow drove the ball home.
St Neots’ Jack Bradshaw went close with a free-kick from just outside the penalty area as the Saints looked to recover from Stour’s two-goal blow.
Green came on with just under 20 minutes to go and looked to make the result secure with a third Stour goal in the 76th minute.
Benbow fooled the hosts with a reverse pass from his free-kick which found Green on the left-hand side of the box. The sub did brilliantly and his rising shot went into the top corner of the net.
Meechan hit the bar for St Neots with a rasping drive in the 82nd minute and Westlake then fired over for Stour after beating two men.
Recent loan signing Will Marsh came on for his Glassboys debut with five minutes left and the hosts made things interesting late on as Bradshaw headed home from close range in the 86th minute after Stour failed to clear a free-kick from the right.
Stour subsequently held on in a nervous finish to book their place in the FA Trophy Third Round Qualifying.
The draw for the final qualifying round will be made on Monday afternoon and is usually released around 1pm.
Stourbridge: Gould, Christophorou, Westlake, McCone, Pierpoint, Forde, Tonks, Broadhurst, Perry (Duggan 87), Benbow (Marsh 85), Shearer (Green 73). Subs not used: Cooke.
Video highlights - https://www.youtube.com/watch?v=dWMZQ0fDdXo
Our next fixture is away at Shaw Lane on Saturday, 18th November. Kick-off will be at 3.00 pm.
YOUTH TEAM & ACADEMY
The Youth Team travel to AFC Wulfrunians next Wednesday, 15th November, in the Joel Richards Memorial Cup (Birmingham County Youth Cup). Kick-off is at 7.45 pm.
The Academy team are at home on Wednesday, 22nd November, when the visitors will be Romulus. Kick-off will be at 2.00 pm at the War Memorial Ground.
The Glassgirls travel to Coventry Development on Sunday, 12th November, for a WMWL fixture. Kick off is at 2.00 pm at Sphinx Sports & Social Club, Sphinx Drive, Coventry CV3 1WA.
RESULTS – SATURDAY 11TH NOVEMBER
BUILDBASE FA TROPHY – SECOND QUALIFYING ROUND
(ties involving Evo-Stik Premier clubs, and others of interest)
Altrincham 4-1 Ramsbottom United
Alvechurch 1-5 Coalville Town
Atherton Collieries 1-5 Marine
Barwell 0-1 Mildenhall Town
Chasetown 1-3 Workington
Hednesford Town 1-1 Cleethorpes Town
Hereford 0-0 Potters Bar Town
Kidsgrove Athletic 0-2 Grantham Town
Lancaster City 3-0 Stratford Town
St Neots Town 2-3 STOURBRIDGE
Stafford Rangers 3-1 South Shields
Stalybridge Celtic 3-2 Rushall Olympic
Warrington Town 1-1 Ashton United
Buxton 3-3 Farsley Celtic
Halesowen Town 0-0 Shaw Lane
Matlock Town 0-3 Nantwich Town
Mickleover Sports 2-0 Sutton Coldfield Town
Updated 17:04 - 15 Nov 2017 by Nigel Gregg