The big striker's third goal in two matches decided a hard-fought encounter on a wet, blustery afternoon, lifting Simon Weaver's team one place to sixth in the National League standings.
Town had the better of the first period, creating a number of promising openings with deliveries into the penalty area, but without really troubling visiting stopper Carl Pentney.
The one time that the Magpies' number one was seriously extended was in the eighth minute when he dived full-length to his right to tip Josh Falkingham's low drive around his post.
Kelvin Langmead was twice unable to direct close-range headers on target when well-placed, either side of Jack Emmett seeing a volley blocked and then lifting over the top from Ryan Fallowfield's right-wing centre.
The closing stages of the opening half saw the hosts produce some slick football, and although the second began in rather scrappy fashion, Town moved ahead eight minutes after the resumption.
George Thomson lofted a left-wing cross to the far post where Beck rose and sent a powerful downward header back across Pentney into the back of the net, despite the best efforts of a defender on the goal-line.
Moments later, Jack Muldoon made an intelligent run to the near post to meet Thomson's free-kick from the right, however Pentney was able to make a routine stop.
Harrogate were then indebted to their own gloveman, James Belshaw, as he denied United an equaliser with an incredible 64th-minute save.
Ryan Upward's header from an inviting free-kick looked destined for the top corner, only for the home custodian to fling himself into the air and claw the ball onto the underside of his cross-bar.
The closing stages saw Beck and Muldoon both off-target with difficult headed opportunities as Town sought to put the game to bed.
Yet, in the end, one goal proved to be enough as Weaver's men chalked up their fourth win in five.