Ched Evans' goal just before the break gave the Blades a victory in their first outing since Speed left to take over as Wales boss.
Caretaker manager John Carver said: "I'm delighted for the boys. They deserved that.
"It's not been the easiest week with the manager going but they've responded in exactly the right fashion, exactly the way that everyone would want.
"I want to say thank you to Gary because he rang me this morning and said he believed in this team and thank you to the players because they stuck completely to the game plan absolutely to the letter.
"And also thank you to the fans because I know what it's like being a supporter sat in the stands when your team had been struggling. But I think they stuck with us and, as the game developed, started to see what we were trying to do."
Source: PA
Source: PA