Sam Johnstone produced a wonderful fingertip save to turn away a snapshot from Altrincham defender James Smith, before Brady opened the scoring with a 25-yard rocket just before the half-hour mark.
Ole Gunnar Solskjaer and Warren Joyce, who included trialists Louis and Sam Barnes in the full-back positions, introduced Conor Devlin, Scott Wootton and Rodrigo Possebon at the interval.
The Reds pressed for a second goal, but were thwarted twice in quick succession when home goalkeeper Stuart Coburn parried efforts from Possebon and Dudgeon.
The latter had the last laugh, however, when he bagged the killer second goal, 15 minutes from time. Will Keane's shot thumped against the underside of the crossbar, and Dudgeon was on hand to convert the rebound.
United: Johnstone (Devlin 46); S Barnes, Cathcart, Gill (Wootton 46), L Barnes (Dudgeon 65); Brady, Norwood, Morrison (Lingard 70), Eikrem (Possebon 46), Ajose (Fryers 46); W Keane.