A first-half goal from Mikel Arteta and a second-half strike by Yakubu were all the visitors needed to secure the points against a team that hardly managed a meaningful response.
Tim Howard was barely tested in the 90 minutes and Derby could have no complaints about a defeat that leaves them stuck in the bottom three.
Tim Cahill also put a header wide from a cross by former Derby midfielder Lee Carsley, but the home side's good fortune ran out in the 25th minute.
The only work that Tim Howard had to do before the break was to drop onto a 25-yard shot from Stephen Pearson but the chances of a more competitive second half were quickly ended as Everton took control from the start.
With Leon Osman outstanding in midfield, Everton did much as they pleased and it was no surprise when they scored a second goal in the 63rd minute.
Osman's pass after he cut in from the left opened up Derby once again and Yakubu took one touch before firing low past Bywater and inside the left-hand post.
The Rams tried to inject some purpose into their play by bringing on record signing Robert Earnshaw and Steve Howard but Everton were rarely troubled.
Alan Stubbs replaced the injured Leighton Baines at half-time and Joseph Yobo dealt comfortably with the few attacks Derby mustered and it was the visitors who always looked more likely to score.
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