Mitch Curry's 87th minute finish sent Town home with all three points after what was looking like turning into a frustrating afternoon at Tameside Stadium.
Joe Leesley's ninth of the season gave Town the lead, but Curzon hit back with a super strike from Chris Rowney.
Town had plenty of possession in the second half, but could not find a way through, until the 87th minute when Curry converted Liam Agnew's through ball that sees Town move three points clear at the top of the league.
Simon Weaver made one change from Monday night's FA Cup victory at Bradford Park Avenue as Jack Emmett returned to the starting XI.
Both sides had claims for a penalty in the opening ten minutes, but both appeals were unsuccessful as Referee Anthony Tankard waved play on.
Home Captain Joe Guest was first to go down after creeping past Josh Falkingham, before Emmett raced past Daniel Shaw before finding himself on the ground just two minutes later.
With ten minutes gone, Town took the lead as Leesley pulled level with Simon Ainge as the club's top scorer for the season.
Ryan Fallowfield's low cross was only partially cleared; allowing Leesley to collect possession and see his low shot deflect past Cameron Mason.
Curzon were level fifteen minutes later though as, against the run of play, Rowney struck a first time shot into the top corner from the edge of the area.
Town looked for a repsonse, but Leesley's shot could only find the legs of Shaw, moments before the winger's teasin cross was inches too high for Jordan Thewlis' diving header attempt.
Thewlis twice went close to putting his side ahead at the break, but saw his chipped cross scrambled away by Mason before blasting over the bar in first half stoppage time.
Both Warren Burrell and Ryan Fallowfield saw their early attempts miss the target at the start of the first half as Burrell's header missed the post, while Fallowfield's first time strike narrowly evaded the top corner.
Injury forced the withdawal of Ben Parker ten minutes into the half, leaving Falkingham to take the Captain's armband, with his replacement Jack Vann testing Mason soon after entering the action.
Town's second change saw Thewlis replaced by Mitchell Curry, whose headed flick on was collected by Thomson and curled agonizingly wide of the left post.
With three minutes of normal time remaining, Curry won the game for Town, collecting Agnew's through ball and calmly slotting past Mason to seal the win!
Harrogate Town: Belshaw, Fallowfield, Parker (Vann 55), Falkingham, Kennedy, Burrell, Thomson, Emmett, Leesley, Thewlis (Curry 75), Agnew
Subs not used: Cracknell, Middleton, Burdett
Goals: Leesley 10, Curry 87
Curzon Ashton: Mason, Baillie, Guest, Shaw, howson, Regan, Rowney, Marshall, Wharton, Leonard (McKenzie 82), Baker (Wright 85)
Subs not used: Clark, Crothers, Butterworth
Goals: Rowney 25
Booked: Baker, Baillie, Leonard, Shaw
Referee: A Tankard