The League One outfit hit the front courtesy of a Chris Smith own goal before second-half strikes from Kieran Agard and Alex Revell ensured Rotherham's passage through to the Northern Section quarter-finals.
League Two York had their chances but the Millers' quality shone through, according to Evans.
"I thought it was as comfortable a night as we could have wished for. There were a few chances in either box but I thought the gulf in class was enormous," he said.
"They had a couple of chances but we missed a couple as well. You've just got to look at our performance in the second half and you can see how much we've improved in a year.
"I changed the team late Tuesday afternoon when we got to the hotel. I spoke to the chairman and he made me aware that there were more Millers fans travelling and after my comments at the weekend I freshened it up.
"I'm pleased with the way they've worked hard and in the second half our fitness told and we were on top and deserved the result."