Curran struck in the 34th minute with a low shot from the edge of the area which went just inside Nicky Weaver's left-hand post.
Before the goal, Gary Madine twice went close to putting the visitors in front, forcing Weaver to block with his legs early on and then having a header cleared off the line.
Apart from an early spell, during which Chris Sedgwick had a shot deflected wide and a Clinton Morrison effort was blocked, Carlisle dominated the first half.
Curran almost added another early in the second-half with a glancing header which was well saved by Weaver, who got down low to his left.
James Chester then headed just wide as Greg Abbott's men continued to impress.