The Scholars were a whisker away from the opener as Liam Kirton and Butlin almost got on the end of a Jordan Evans whipped cross.
Butlin then had an effort cleared off the line before Kirton collided with the post trying to head home an Oli Hayward ball from the wing.
Chasetown were in the ascendancy and Harrison denied a long range Jack Langston effort by finger tipping the ball over the crossbar.
Kidsgrove should have taken the lead on the stroke of half time. Will Byers crossed from the right and a stretching Ant Malbon some how struck the crossbar from 4 yards.
In the 65th minute, Chasetown were awarded a spot kick when Josh Thorpe clattered Kirton. Harrison saved the penalty to his left but Langston followed up to head home.
Harrison then handled outside the area and when Butlin's free kick was blocked, Langston fired the rebound wide.
Chasetown were reduced to ten men when Lewis Riley-Stewart received a second yellow card.
Athletic hit the bar again in the 85th minute when a right footed shot from Kingsley Adu-Gyamfi was flicked onto the frame of the goal by Curtis Pond.
Chase substitute Josh Cooke almost wrapped it up in stoppage time when his low shot struck the base of the post.
Kidsgrove sent everyone forward including their goalkeeper but Chasetown saw out the victory.
Elsewhere, OJM Black Country lost 1-0 to Matlock Town, Sporting Khalsa lost 2-1 to Leek Town and Walsall Wood also fell to defeat to Worcester City FC.
Tividale did claim a 1-0 win against Market Drayton Town, courtesy of a Dan Smith strike.