Lee Novak set up two goals in 66 seconds as Huddersfield made it to the fourth round for only the second time in 11 years.
Terriers striker Novak raced to the by-line in the seventh minute and pulled the ball across the face of goal for Arfield to tap in the opener.
Novak then latched on to a long ball out of defence down the right flank virtually straight from the re-start and crossed for Roberts to blast in at the far post.
Town sat back after that and nearly let the Blue Square South side back in it. Town skipper Peter Clarke blocked a goal-bound header from Olly Schulz in the 24th minute and from the resulting corner Roberts had to clear off the line another Schulz header.
The Whites, backed by an impressive 1,535 travelling fans, also went close when Town goalkeeper Alex Smithies made a mess of dealing with a long ball when he slipped before recovering to deny 29-goal top scorer Adam Birchall a tap-in.
Town had chances to extend their lead after the break. Roberts and Kevin Kilbane both had shots parried by Dover goalkeeper Ross Flitney.