HALIFAX eased into a fifth round trip to Nuneaton Borough with an impressive second-half display capped by a fine goal from Jon Shaw.
The Shaymen's leading scorer grabbed his 11th of the season when he headed home Ryan Toulson's cross in the 64th minute.
Halifax could have won more convincingly had they taken any of the other good chances they created against their high-flying Blue Square Premier hosts.
Halifax took the lead after 18 minutes when Steve Torpey, the outstanding player, had a 20-yard free-kick deflect past Danny Naisbitt.
The home side hit back seven minutes later when Antonio Murray nodded home from close range after Nathaniel Knight-Percival had flicked on Gareth Gwilim's long throw.
Nicky Gray volleyed off target and Murray sent another header over as the first half closed.
After the break it was all Halifax as they pinned their hosts inside their own half. Shaw had a 54th-minute effort blocked and then ran through on goal only to be denied by the goalkeeper.
Substitute Daryl Taylor lifted a close-range effort over on the hour, but the breakthrough came four minutes later when Shaw stole in at the near post to meet Toulson's delivery.
Torpey, Griffith and Stamp all had chances but found Naisbitt in good form and there was a late scare when Adrian Cambridge shot from long range but it was deflected wide.
Histon: Naisbitt; Okay, Mitchell-King, Langston, Kennedy; Barker, Knight-Percival (Cambridge) 46, Gwillim, Murray (Dillon 52), Wright, Pope (Ada 54). Unused substitutes: Key, Nightingale.
Halifax Town: Mawson; Toulson, Scott, Wright, Young; Griffith, Heslop, Gray (Taylor 51), Torpey; Stamp, Shaw. Unused substitutes: Legzdins, Atherton, Killeen.
Referee: G Stratton (Leicestershire).
Halifax man of the match: Steve Torpey.