Harriers started the second half on the attack and Broughton drilled a low shot at the well positioned Book from just inside the area.
Broughton was then booked after a foul to the delight of the home crowd.
Cheltenham now appeared to be calling the shots and Grayson put them back into the game on 54 minutes.
The experienced forward glanced home a left-wing cross from strike partner MacDonald with keeper Clarke left terribly exposed.
Harrier fought their way back into the game with Doyle lively on the left hand side without giving Book any problems.
Harriers grabbed a crucial third goal on 64 minutes when Doyle's mesmerising left-wing run took him past two players and into the penalty area. Daire Doyle
The Irish midfielder clipped a fine cross for Broughton to head home at the far post despite Book's valiant effort to keep the ball out. Harriers Stamps was booked on 69 minutes after a crunching tackle on substitute McAuley who replaced McCann three minutes earlier.
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