The champions were struggling to break down their obdurate visitors until Arfield, returning from injury as a second half substitute, scored the only goal of the game in the 75th minute.
Van Bronckhorst, who was also able to welcome Ryan Jack back into his squad as a late sub, was gratified by his team’s persistence on a night when the absent quartet of Alfredo Morelos, Ianis Hagi, Joe Aribo and Ryan Kent was keenly felt.
“I’m very proud because it was a difficult game, in very difficult circumstances,” said van Bronckhorst.
“If you see the chances we created in the first half and then in the second half, if you don’t score it’s going to be difficult no matter what because time ticks on.
“But I was happy with the subs we could use. Scott has been out for a while but he could play the last stage of the game and he changed it.
“I’m really pleased for Scott and the team because in the circumstances we still kept fighting to win the game. Three points were big.
“It was good to have Scott back and to be able to put him on. The same for Kemar Roofe and Ryan Jack as well. It’s looking good because the injured players are coming back. For the weekend against Ross County in Dingwall, we will also have Joe Aribo and Ryan Kent back.”
Van Bronckhorst admitted his team lacked tempo and precision against a Livingston side who defended in depth but felt the victory was ultimately fully merited.
“It’s not easy against the low block because you need to be really accurate with your passing and movement,” he said. “The speed of play we wanted, it was very difficult because of the circumstances.
“That’s why we were struggling a little bit to find the spaces. But I think we created enough chances to win the game, so I was really pleased when one went in.”