The second-placed Saints went into the game trailing the leaders by three points but ended the day level with them after a 1-0 victory and yet another clean sheet for Kevin Wilkin’s men.
The game, which attracted a big attendance of 1,557, was delayed during the first half after a spectator was taken ill with the players heading back into the dressing-room for a short time.
When the game resumed, it was a tight and tense affair with Shepherd Murombedzi and Matt Lowe both going close for the hosts.
But the decisive moment came with 10 minutes to go when Brackley were awarded a penalty and James Armson stepped up to convert from the spot.
Brackley can move to the top if they avoid defeat in their game in hand on Gateshead at Curzon Ashton on Tuesday night.