Anton Dowds pounced on Craig Watson's loose header back to goalkeeper Robbie Hemfrey to slot in the hosts' equaliser just after half-time.
Ayr quickly moved ahead when Sean McGinty headed in substitute Jamie Murphy's free-kick.
Ayr Untied captain Sean McGinty: 'It's fantastic. We've not been very good at home this season. It's a massive night for the football club.'
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