The Nigerian returned to form with a brilliant dummy to set up Mikel Arteta's first-half opener then coolly slotted away the decisive second with 62 minutes on the clock.
The goal was only Yakubu's second Barclays Premier League strike since his big-money move from Middlesbrough but Moyes insists there is so much more to his play.
"Today he delivered with an impressive goal that took the pressure off us and a great dummy. He showed intelligence and awareness and he is a better footballer than I thought.
"But in Britain what we really enjoy is seeing players put in their graft. That's part of our culture and Yakubu did that today and he got a standing ovation for us."
Moyes admitted he was relieved to see his side get back to winning ways after two successive defeats and was pleased with his defence who never looked likely to be breached by a lightweight Rams attack.
Moyes added: "We didn't really look under a great deal of pressure today. We knew it would be a tough game because Derby have a good record at home but for long periods we had it under control."
Rams boss Billy Davies admitted his side fully deserved to lose to a better side but vowed to bridge the quality gap by bringing in plenty of fresh faces in the new year transfer window.
Davies said: "We've got no excuses today. We got beaten by the better side who showed much more quality than we showed, and in the end they fully deserved their victory.
"We've just got to do what we can with what we've got and the facts of the matter are teams we're playing at the moment just have that little edge over us in terms of quality.
"We just have to keep working very very hard to get to the next transfer window and add in many areas of the side. I said before the season kicked off the second window was going to be crucial for us.
"All I'll continue to do is ask these players to keep giving the best that they've got. Their attitude and their spirit in the dressing room is never in question and today we just didn't quite fire on all cylinders."