York won their first round replay 3-0, thanks to two goals from Matty Blair and one from Chris Smith.
"From start to finish, the desire to win the game was important," Mills told BBC Radio York.
"We showed desire from the moment we turned up to play this game. The performance was excellent."
York were much improved from Saturday, when they needed Jon Challinor's late equaliser to earn a 2-2 draw with the Moors at Bootham Crescent.
"We wanted to put it right from Saturday," said Mills. "We knew the standard of our game wasn't what it should be on Saturday.
"We made a couple of changes and changed a few things round and it's worked well for us.
"Everybody that's played, everybody that's turned up on the bus tonight has wanted to win.
"Three goals away from home, a clean sheet and through to the next round. I'm delighted for everybody at the football club. It's a great night for us."
York were beaten semi-finalists in 2008 before going one better a year later when Martin Foyle's side got to Wembley, where they lost 2-0 to Stevenage.
But, after only making the quarters in 2010, then going out in the first round at home to Boston a year ago, Mills is targeting another run when they head for FA Cup giantkillers Salisbury for their second round tie on 14 January.
"Now we want to go all the way," added Mills. "I've said I want to win it and I think we've proved it tonight. It's a professional performance. An excellent performance."
Source: BBC Sport
Source: FOOTYMAD
