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The Conference Premier's bottom team looked on course to shock the league leaders but Villa netted a free kick in added-on time to deliver a cruel blow to the visitors.
Former Alfreton striker John Akinde was proving to be a handful for his former side, but after failing to strike early, the visitors responded through Dan Bradley, who fired a shot across goal from an acute angle.
The home side were stunned two minutes into the second half when Alfreton took the lead through Massiah McDonald after he directed a header into the corner of the net from close range.
However, Alfreton's lead lasted just 15 minutes as Barnet equalised from the penalty spot. After Lathanial Rowe-Turner had been dismissed for a professional foul, MacDonald slotted the spot kick home to give his team the momentum heading into the final quarter of the match.
Barnet looked like they were about to lose top spot to Halifax Town before Villa fired a free kick past a helpless Dawson to break Alfreton's hearts.