Nicky Clark, playing as a triallist, netted a debut goal after just 30 seconds, while Lewis Macleod was also on target, but it was McCulloch who was singled out for praise by McCoist.
"It was terrific. He just told me that's his first hat-trick," he said.
"I was surprised. I thought he had scored at least another one. He led the line well.
"I don't think I did score a hat-trick of headers. I remember I got two against Vladikavkaz and I probably should have headed the third one in."
Clark scored more than 40 goals for Queen of the South in the same division last season and McCoist was also keen to highlight his contribution.
"I was delighted for him. He has been showing enough in training to indicate that he will score goals," added McCoist.
"He plays on the shoulder, he runs through and he is extremely quick. I think he's got a good range of finishing.
"It was a brilliant start. The wee fellow showed tremendous pace and awareness to get in and finish it.
"It was an unbelievable start after just 30 seconds. It was absolutely ideal. The level of finishing today was excellent."