The Minstermen's 3-0 success over the League Two side was only their second win in 10 games in all competitions.
Mills told BBC Radio York: "We've come out on top and deservedly so. It's a result that will bring a big smile to everyone at the club.
"We have good players here. They've got to start believing in themselves."
Chris Smith's header and a two goals from Michael Rankine, one a penalty, saw City triumph over the Millers at Bootham Crescent on Wednesday.
"I thought Rankine was immense," added Mills. "He's working hard and I've told him that if he keeps doing that then the goals will come.
"(Scoring goals) has been a problem and we're trying to address that. We've had chances in the last couple of games but we haven't put them away.
"The players are well organised and they know what their jobs are. We have been the best side over the two games."
Source: BBC Sport