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.
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.
After the break, Ngog went close to doubling his side's lead but couldn't direct his volley goalwards when well-placed on the edge of the penalty area.
Derby almost got back into the game out of nothing as Paul Coutts fired a vicious effort at goal from range but Adam Bogdan got down quickly turning the ball past the post.
Connor Sammon and Coutts went close again for the Rams before they finally got their equaliser as Jamie Ward headed home Michael Jacobs's cross from close range to earn a point for his side.