The tie eventually finished 1-1 after extra-time, with Michael Coulson's opener for York midway through the second-half soon being cancelled out by Chris Brown from the penalty spot.
Brown was also on target during the shoot-out for the hosts, while Coulson and Matty Blair both blazed over the bar for their opponents as the large bank of York fans watched on at the far end of the stadium.
Bizarrely the decision on the choice of ends had been reached on the advice of the police and not by the toss of a coin.
"The police decided that the penalties had to be taken at that end but that didn't make sense to me," Mills told The Press.
"I thought that would have been decided by a toss-up so I felt a bit aggrieved about it.
"I don't understand why the police have to get involved and, if that was going to be the case, we should have been told before kick off because we worked hard to get through extra-time.
“That’s perhaps a learning curve for the referee but there’s no point in going on about that because we’ve been knocked out now.
"Penalties cost us at the end of the day - they stuck theirs in and two of our players ballooned them over the bar."
The Minstermen will need to swiftly recover from that disappointment as they entertain Wycombe Wanderers in their opening game of the season this weekend.