The Ducks were made to work hard for their win, and had the better of the first half, eventually scoring in the 22nd minute when Vince Melisi's shot went through a crowd of players past Bognor keeper Steve Burton.
Melisi hit the bar on 36 minutes as United put on the pressure before half-time but in a generally disappointing second half Bognor defended stubbornly.
United's only real threat came when Burton saved Steve Ellis's shot at the second attempt, but they held out with keeper Ian Barnes having to make to two good saves in the last few minutes.
David Figg