Visitors Burnham put up a brave fight but holders United eventually ran out easy winners on a bitterly cold night.
Surprisingly, it took United 42 minutes, before Dean Williams headed in a nearpost corner to make it 1-0.
A foul on Adam Clarry just outside the box allowed Kieran Gallagher to strike the free kick home after 49 minutes.
United's fitness was a telling factor, and on 83 minutes Gallagher crossed for Mick Swaysland to head in at the far post, before Williams drove in his second goal on 88 minutes to complete the rout.