Wrexham started the game brightly and perhaps should have gone ahead in the opening salvos - either through Ollie Palmer or Paul Mullin. But against the run of play, veteran striker Muldoon gave the home side the lead with a beautifully guided header in the 24th minute.
George Dobson then spurned a glorious chance from a rapid counter-attack but once again, goalkeeper James Belshaw was equal to the task. The Harrogate stopper then made a reaction save to keep out Jon Dadi Bodvarsson's acrobatic strike from close range.
The League One side looked to have grabbed a late equaliser through substitute Modou Faal but the big striker was offside when he nodded home. Despite piling on the pressure, the Welsh side - who made it to round four for the past two seasons - fell at the first hurdle in the famous competition on Sunday.