That late loss to in-form Boro had been Stoke’s first reverse in seven games, but they had to dig deep to eke out a result in a poor game against an injury-hit Derby side who have no wins in their past six in the Championship.
The only excitement of the first half came as Rams’ striker Carlton Morris threatened to burst through on goal but went down under a challenge from Bosun Lawal, bringing loud protests from Eustace on the sidelines.
Potters boss Mark Robins sent on George Hirst as a substitute at half-time but it was another Scottish summer signing who made the breakthrough just four minutes after the restart.
Ethan Galbraith lifted a free-kick into the box and Graham, signed from Dundee, saw his header strike Morris on the back and loop beyond the grasp of Rams keeper Josh Vickers.
With Lewis Travis limping off shortly afterwards, after making his first start since August due to injury, Derby were cursing their luck.
They offered a response as Rhian Brewster’s shot glanced off the outside of a post, but Stoke countered dangerously, with Hirst, Djibril Soumare and Justin Devenny all going close to adding a second.
Derby summoned one last effort and almost snatched a point as Morris headed the ball on and Ozoh’s volley was heading for the bottom corner until Johansson got down brilliantly to turn it away.