The Glassboys had survived two brushes with the woodwork in the first half and just as the game threatened to peter out, substitute Marcus Marshall’s fine individual effort decided this first qualifying round tie.
Marshall jinked past one defender and then surged past another before his deflected shot made its way past Charlie Price 12 minutes from time and just six minutes after coming on.
Basford almost scored in the opening two minutes when Brad Gascoigne hit the bar with a free header from Ryan Wilson’s corner and Price was called into action in the 20th minute, parrying wide a Wilson shot following good work from Alex Howes and Terry Hawkridge.
Owen Betts then rattled the bar with Price beaten nine minutes before half-time, while Stourbridge’s best opening of the half came through James McQuilkin’s weaving run and shot that was gathered by Adam Collin.
The best chance the visitors had came five minutes after half-time when Montel Gibson lifted over with the goal at his mercy.
There was a late moment of hope for Stour but Nick Clayton-Phillips had his goalbound effort blocked by Declan Dunn with six minutes to go.