Bolton Wanderers lead at half time at Pride Park thanks to defender Craig Dawson's 45th-minute header.
The Trotters were first to have a crack on goal as the ball fell to David Ngog 20 yards out but his effort was tame and was easily saved by Adam Legzdins.
Derby were struggling to get any forward momentum but almost took a surprise lead when Jeff Hendrick found Craig Bryson in the area, but the midfielder could only head wide.
This chance seemed to inspire the hosts as they came close again through Paul Coutts but he couldn't find the target when well-placed in the box.
Despite Derby's strong end to the half it was Bolton who took the lead as Dawson headed home from close range from Jay Spearing's free kick.
