Chesterfield dominated play but could not find a way past a stubborn and well organised away defence.
Town goalkeeper Kyle Letheren was the hero in the end with two saves in the shootout.
The Blues picked up £4,500 in prize money for winning the the tie.
The draw for the next round is at 1pm on Monday.
Here are Liam Norcliffe’s player ratings from the game...
1. Kyle Letheren 8
Four clean sheets in a row for Letheren and just one goal conceded in the last five games. Saved two penalties in the shootout. Distribution was good throughout the game. Didn't really have a safe to make as Brackley failed to threaten. Photo: Dean Atkins
2. Josef Yarney 6
Kept his place in the side after last week's big win agains Barnet and he was steady again, this time at right wing-back. Replaced by Joel Taylor on the hour. Photo: Steve Flynn/AHPIX LTD
3. Will Evans 7
Slotted back into the right of a three-man defence after missing out through suspension last weekend. He stepped out from the back a lot in the first half and his diagonal passes in the air to Carline caused Brackley a lot of problems. Did not see as much of him in an attacking sense after the break although he almost scored from Taylor's cross. Photo: Andrew Roe/AHPIX LTD
4. Haydn Hollis 8
Hollis has been at the heart of the defence which is in fine form at the moment. He was calm on the ball and tidied everything up at the back. Another good performance from him. Photo: Rachel Atkins