Stuart Elliott nearly got his first Boothferry Park goal at the start of the second half, but his volley went inches wide.
However, Dudfield was on hand again to make it 2-0 just minutes later. Mike Edwards played a ball through the middle which Gary Alexander let run on, and Dudfield raced clear of the defence, went round the keeper and tapped in for his second.
City nearly got a third when Alexander's cross found substitute Mark Greaves, who headed across goal needing just a touch from Dudfield or Williams to redirect it into the net. Neither could quite get there and the ball bounced inches wide of the post.
Phil Jevons got one back in the dying minutes with a header, but the game was already dead with the constant substitutions, and City held on for the 2-1 win.
City line-up (4-3-3): Glennon; Edwards, Anderson, Strong, Smith; Williams (Itonga), Ashbee, Appleby (Price); Dudfield (Peat), Alexander, Elliott (Greaves).
Un-used subs: Musselwhite, Whittle, Philpott.