Ashton kept out Elliott Bennett's spot-kick deep in stoppage-time to ensure the Evo-Stik Premier Division outfit will be in the draw for the third round.
The visitors shocked the League One leaders when midfielder Nicky Platt drove them ahead six minutes before the break.
FC United had defender Scott McManus sent off for an elbow on substitute Gary Hart in the 69th minute and held out until Mauricio Taricco levelled with a header which went in via a deflection off substitute Francisco Sandaza.
Brighton could have won it in the fifth minute of stoppage time when Karl Munroe brought down Sandaza, but Bennett's spot-kick was brilliantly saved by keeper Ashton diving to his right.
Ashton played a vital role in his side's gallant display by pulling off four superb saves before Platt's opener.