They partly had Charlie Price to thank for making three great saves, including a point blank one from Dan Sweeney in injury time.
Redditch looked the most likely to go in at the break ahead. Price saved a Sweeney shot with his feet on 10 minutes and the same player then hit the post from a free kick on 21 minutes.
Good work by Alex Cameron set up Sweeney on 38 and his shot was again well saved by Price to keep the score at 0-0 at half time, with Stourbridge not really threatening at all.
But on 48 minutes a shot by Ben O'Hanlon deflected into space and Styche took advantage, making no mistake from inside the box.
Styche had a chance to make it two five minutes from time but tried to lob goalkeeper Kieran Boucher who easily grabbed the shot.
Stourbridge will now look forward to the first round draw on Monday.