With goal mouth scrambles a regular occurence, neither side could find the crucial finishing touch that would have secured all three points.
Simon Weaver made one change from Tuesday night's clash with Spennymoor Town as Jake Wright replaced Dom Knowles in attack.
The contest started at a frantic pace as Jack Emmett sliced John McCombe's knock back just off target, moments before the centre back was well placed to block a strong attempt on goal.
Glenn Walker and Emmett then both went close with individual efforts taking on players, but shooting just off target.
The rain had been hammering down all afternoon, resulting in large areas of mud around the field, with one near the Brackley goalmouth contributing to a scramble eventually cleared after good work from Wright to get to the byline.
Town almost edged ahead in the 37th minute when Joe Leesley's curling effort was dropped by Danny Lewis, but the Brackley stopper recovered in time to keep out Wright's follow up.
Four minutes later James Armson found space on the edge of the box and tried to pick out the bottom corner, but James Belshaw kept out the effort with a fantastic one handed save.
On the hour mark, moments after Emmett's pull back had agonizingly missed everyone in the box, Lee Ndlovu could only find the side netting when trying to convert Shane Byrne's low cross.
With 16 minutes of normal time remaining Belshaw made a crucial save to keep the score level, keeping out Aaron Williams' finish when the Brackley top scorer had a clear sight of goal.
Ndlovu almost found the winning touch with five minutes to play, but the ball somehow squirmed wide of the post as the home number nine got on the end of a low cross at the back post.
A minute later home fans were incensed as Belshaw looked to bring down a Brackley attacker in the box, but the Town stopper won the ball first and ensured the points were shared.
Harrogate Town: Belshaw, Burrell, Parker, Falkingham, Kitching, McCombe, Thomson, Emmett, Beck, Wright, Leesley (Thewlis 75)
Subs not used: Knowles, Cracknell, Fallowfield, Agnew
Brackley: Lewis, Murombedzi, Lowe, Byrne, Graham, Dean, Walker, Armson (Nicholson 69), Ndlovu, Williams (Iaciofano 81), Walker
Subs not used: Brown, Streete, Myles
Referee: P Howard