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Wimborne had an early penalty appeal waved away when striker Cameron Munn felt he was brought down in the box.
The Dorset part-timers took the lead after only nine minutes when property surveyor Yang Seung-woo curled a superb strike from just inside the penalty area into the top corner.
Crawley came close to equalising when Kai Jennings flashed a low shot narrowly wide before Jonny Russell headed tamely into the arms of home goalkeeper Cam Plain.
The visitors’ former Arsenal goalkeeper Vito Mannone made a fine save to deny Wimborne’s Joe Smith a second goal as he raced clean through.
Crawley’s interim manager Kevin Blackwell made a triple change at half-time and Charlie Barker fired inches wide for the team 57 places above their hosts in the pyramid.
Southern Premier League Wimborne got a huge slice of luck to seal victory 11 minutes from time. Jackson King’s shot hit Munn to send the ball squirming beyond Mannone.
It is the second season in a row that Crawley have been knocked out by non-League opponents after losing to Boreham Wood last season.