Cameron Jerome, making his first start of the season, scored after just 30 seconds and then, following Matt Oakley's second-half equaliser, scored the winner just after the hour.
Bruce was delighted with the display of Jerome and strike partner Mikael Forssell, who could have had at least two himself before being replaced in the second half.
He said: "You don't get any better a start than that. Fantastic.
"We made hard work of it but the only thing you can do is make chances and we should have been four up at half-time and the game should have been out of sight.
"You are sitting there fearing what might happen - and what did happen. They haven't created anything at all against us and we have missed chance after chance and we haven't taken them.
"We needed just to have taken the chances and that would have been a complete performance because it could have been a rout.
"Mikael Forssell could have had four. I thought he and Cameron were immense and the two of them gave Derby a torrid time.
"It was certainly one of Cameron's best performances - and at this level - and he was excellent for us."