Victory also extended Burnley's lead at the top of the Sky Bet Championship following QPR's draw at Millwall.
"It is all about small margins. They had the better of the first half but we had a solid framework, did the basics and were solid enough not to concede," Dyche said.
"The second half levelled out, although there were pockets of good play. I was thinking it was going to be a 0-0, and then a fine moment of quality saw a good cross and a good header."
Dyche was delighted with Arfield's contribution after deadly front duo Sam Vokes and Danny Ings endured a difficult afternoon.
"I think both pairs of centre-forwards are all very good but defences were on top," he said.
"We had to find a way to overcome that and we did with some very good play."