Huddersfield midfielder Gary Roberts was shown a straight red card in the 37th minute after a strong challenge on Darlington's Ricky Ravenhill saw the combative midfielder stretchered off.
The visitors had been second best to the League Two hosts who had earlier gone close through David Poole in the 18th minute and then Liechtenstein international Franz Burgmeier in the 34th minute.
And 10-man Town went a goal behind on the stroke of half-time when Burgmeier's free-kick was met by the head of defender White, who powered in from 10 yards.
Huddersfield substitute Andy Butler went close to an equaliser in the 52nd minute when he headed against the Darlington crossbar.
The visitors pressed for an equaliser and striker Keigan Parker was inches wide with a right-foot drive from the edge of the area in the 67th minute, while Darlington captain Steve Foster brought a fine save from Alex Smithies with 13 minutes left as the hosts progressed to the next round.