The game looked to be heading to a goalless draw until Beck popped up with the winner.
Halifax will feel they deserved at least a point from the game having made all the running in the second-half, but were beaten at home for only the second time this season.
Halifax grew into the game after a slow start, with Harrogate goalkeeper James Belshaw keeping out a fierce low shot by Charlie Cooper and a Liam McAlinden free-kick.
But Warren Burrell’s ferocious shot which cracked off the crossbar and Ryan Fallowfield’s disallowed goal for offside were warnings that the hosts needed to heed.
Beck also went close for Harrogate with a header that Sam Johnson tipped behind, with half-time coming at a good time for Halifax as the visitors began to dominate.
There were flashes of inspiration from Michael Duckworth, who saw a shot blocked and a long-range strike that Belshaw turned behind.
Substitutes Dayle Southwell and Cameron King then combined as the latter saw a terrific effort whistle narrowly over.
The pressure on Harrogate was building, with Southwell’s shot deflected behind before Belshaw failed to claim a corner, requiring two brave blocks to keep out Halifax shots in the aftermath.
But then came the sucker-punch as Beck poked in at the near post after a low right-wing cross was not cleared.
Halifax: Johnson, Duckworth, Clarke, Staunton, Binnom-Williams, Nolan, Cooper, J King (C King 67), McAlinden, Williams (Southwell 72), Sho-Silva. Subs not used: Appleyard, Hanson, Earing.
Harrogate: Belshaw, Fallowfield, G Smith, Hall, W Smith, Burrell, Bradley (Brown 80), Thomson, Kiernan Diamond 70), Muldoon, Beck. Unused substitutes: Stead, Emmett, Cracknell.
Referee: S Mather (England).