Millwall opened the scoring five minutes after the break when Moses Ashikodi found Sweeney on th left, who then dispatched a low, left-foot curler past Paul Hyde in the Dover goal.
Dover grabbed an equaliser ten minutes from time when Issy Armadi broke into the area and netted after seeing his first effort saved by Millwall keeper Masterson.
But with just three minutes left, The Lions grabbed the winner when Sweeney latched onto a loose ball and fired it homw from 20 yards out.
Millwall: Masterson, Dunne, Bull (Donovan, 86mins), M.Elliott, Rees, Clancy, Cogan, Hearn, Ashikodi, Braniff (Quigley, 75mins), Sweeney. Subs not used: Overland, A.Robinson, T.Robinson.