The Shaymen overcame the League Two outfit thanks to a goal from Cameron King early in the first-half to put his side into the Second Round for the first time in a decade.
They came under inevitable pressure in the second-half but Fullarton said he was pleased with how his side dealt with it.
“I thought it was almost a complete game from us," he told the club's official website.
"What I mean by that is, in the first-half, with the ball and possession, I thought we showed guile, understanding and created a number of chances.
"But then obviously the second-half we had to defend resolutely. We were resilient and had to put our bodies on the line and it resulted in us getting the 1-0 victory.”
"I think our defensive shape as a team was fantastic. It was called upon second-half and it was needed because at times we were under severe pressure.
"But the determination throughout was really pleasing to see.”
FC Halifax Town will now play AFC Wimbledon in the Second Round on December 1.