Louis Almond gave the hosts the lead from the penalty spot after keeper Dean Snedker brought down Mike Phenix but James McQuilkin fired in an equaliser.
Elton Ngwatala put Kiddy ahead at the break but Phenix levelled prior to Tyrone Williams giving the visitors the lead once more.
Evan Garnett made it 4-2 to Harriers but Jack Ryan pulled a late goal back.
Attendance: 1,473
Referee: Derek Eaton
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