Luke Rodgers put Port Vale in front in their first tie but Chasetown would not be denied and Mark Branch equalised with a 40-yard free-kick.
In the replay at the Scholars Ground Rodgers missed two penalties for Vale and the match looked to be heading for extra-time before the decisive 89th-minute winner.
Chasetown substitute Smith struck with a header, much to the delight of the 1,986 packed inside the ground. Mass celebrations followed the final whistle.
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