Lindon Meikle's deflected strike was enough to book Eastwood's place in the second round of the FA Cup.
The hosts started brightly, but a fine 25-yard strike from Tom Winters past keeper Shane Redmond gave the visitors a half-time lead.
Eastwood hit right back with Russell Cooke powerfully heading in Richard Dunning's near-post corner.
It was left for Meikle to complete the turnaround when his shot was helped in off defender David Perpetuini.
Eastwood Town: Redmond, Asher, Cooke, Sturdy, Robinson, Meikle (Diuk 87), Foster, Dunning, Todd, Gardner (Rhead 50), Holmes.
Subs Not Used: Smith, Knox, Whitman, Cooper, Deakin.
Goals: Cooke 54, Meikle 65.
Brackley: Knight, Fontanelle, Pertetunia (Patterson 72), Green, Blossom, Farley (Harper 63), Pierson, Cracknell, Thorpe (Spencer 34), Mackey, Winters.
Subs Not Used: Garcia, Longford, Willshere.
Booked: Patterson.
Goals: Winters 20.
Att: 960
Ref: Patrick Quinn (Middlesborough).
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