The experienced midfielder's first-half header at the iPro Stadium ensured Steve McClaren's fourth-placed side closed the gap on their opponents to just a point.
On-loan forward Patrick Bamford had an early chance to open the scoring when he raced clear but took too long to get his shot away and Richard Dunne intervened.
Derby were the better side in the first half and Eustace ended his goal drought after 20 minutes with a header that looped over Rob Green after the former England goalkeeper could only palm away a Will Hughes corner.
Lee Grant did well to keep out Aaron Hughes header before half-time and Harry Redknapp's side had the upper hand in the second half, but were toothless in the absence of top scorer Charlie Austin and Derby dug deep to secure three precious points.
Rangers missed the chance to go back above Burnley into second spot as Derby enhanced their chances of securing automatic promotion.
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