The Saints were beaten 3-0 at Blyth Spartans on Saturday and knew three points were needed against North Ferriby on Tuesday night.
The St James Park pitch passed a late inspection and Brackley took full advantage to record a comfortable 3-0 victory.
Wilkin’s men remain unbeaten at home in the League this season and they’re now six points adrift of leaders Salford City ahead of Saturday’s game against resurgent Darlington.
“Everyone was mightily disappointed after Saturday’s performance so it was good to have another game again so soon,” Wilkin told Brackley’s website.
“We started the game with good purpose and showed what we are about. Ferriby never gave up and we had to manage the game. We showed good patience and took our chances.
“It is good to get a bit of confidence back ahead of what will be a very tough game when Darlington come here on Saturday.”
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