Goals from Mikel Arteta and Yakubu gave Everton victory and sent Derby back to the bottom of the Premier League.
Everton took the lead at Pride Park when Arteta beat goalkeeper Stephen Bywater from 10 yards after Yakubu dummied Phil Jagielka's long ball.
Yakubu made it 2-0 after the break when he latched on to Steven Pienaar's ball to fire in from close range.
Derby's best chance came from the boot of Stephen Pearson whose 25-yard effort was comfortably saved by Tim Howard.
The result was fair but Everton's win owed more to Derby's lack of discipline than their own attacking endeavours.
606: DEBATE
The Rams, who had not scored for more than four hours in the league, began brightly with some delightful exchanges taking place on their right between James McEveley and Eddie Lewis.
But for all their good intentions Billy Davies' men lacked the guile to cut through a fairly sturdy Everton defence.
The blue brigade from Merseyside were hardly Brazil today, but the move that led to the first goal would not have looked out of place at the Maracana.
Jagielka's long, low pass from the right-back slot was directed at Yakubu, but the crafty Nigerian let the ball run under his legs knowing that Arteta was behind him.
The Spaniard looked up once before firing a firm shot into Bywater's right-hand corner, despite the keeper managing to get half a glove on the ball.
The goal knocked the stuffing out of the Rams who started to play like a side fancied by many as favourites to go down this season.
Davies introduced strikers Robert Earnshaw and Steve Howard after the break, but just as they were looking to assert their influence Everton grabbed their second.
South Africa midfielder Pienaar, who was probably the man-of-the-match because of his sensational workrate, won the ball on the right and poked a pass in the direction of Tim Cahill and Yakubu.
This time it was Australian Cahill who played a subtle dummy and let the ball run through to his burly team-mate who fired past the outstretched leg of Bywater from just inside the six-yard area.
Derby still had 20 minutes to get back into the game but their best effort of the match had come earlier on when Pearson managed to hit the target but failed to attach any power to his strike.
Derby boss Billy Davies:
"The facts of the matter are teams we're playing 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."
Everton boss David Moyes:
"Today he delivered with an impressive goal that took the pressure off us and a great dummy. He showed intelligence and awareness.
"But in Britain what we enjoy is seeing players put in their graft. That's part of our culture and Yakubu did that and got a standing ovation for us."
Derby: Bywater, Mears, Davis, Moore, McEveley, Fagan (Howard 62), Oakley, Pearson, Lewis (Earnshaw 57), Miller (Jones 80), Barnes.
Subs Not Used: Price, Leacock.
Booked: Moore.
Everton: Howard, Jagielka, Lescott, Yobo, Baines (Stubbs 46), Arteta, Osman, Cahill (McFadden 86), Carsley, Pienaar, Yakubu (Anichebe 81).
Subs Not Used: Wessels, Gravesen.
Booked: Jagielka.
Goals: Arteta 26, Yakubu 63.
Att: 33,048
Ref: Steve Bennett (Kent).
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