They proved their determination after only a minute when a mix-up between home keeper de how Ed de Goey and John Halls allowed Lee Morris to nip at the back post and toe poke the ball home into the bottom corner from just five yards.
After being camped in the Rams half, Stoke finally got their reward just before the half hour when de Goey's long free-kick bounced over the Derby defence to connect with Ade Akinbiyi who struck first time and buried the ball into Lee Grant's goal.
Morris warned City's defence not to fall asleep and almost grabbed his second when he hit a roaring shot from 25 yards that left de Goey stranded, but cleared the cross bar.
The travelling fans were silenced in the 53rd minute though when John Eustace whipped in a free-kick for a crowd of heads at the far post, but it was on loan Gerry Taggart who leaped the highest and powered the ball straight thorough Grant to give City the lead.
Wayne Thomas almost equalised for the away side ten minutes later when a Bolder cross zipped across the box and an attempt to slide the ball out resulted in the heart and mouth moment for the Britannia crowd as it fell on to the post.
A Derby double substitution seemed to help slightly and they did come close to equalising minutes from the end, but Manel's header wasn't quite good enough with the ball bouncing off the frame yet again.