The Millers, a division below the Magpies, coasted into the third round where they will travel to Aldershot next month.
First-half goals from Ben Pringle, Mark Bradley and Daniel Nardiello secured a deserved 3-0 victory.
It was a miserable night for County whose poor home form continued and they saw defender Damion Stewart sent off after half an hour for a deliberate elbow on Alex Revell.
Curle, who locked himself in the dressing room with the players for almost an hour after the final whistle, said: "First of all I look at myself and I look at my staff.
"I go back to when we first came in and what we brought to the players and that is what I've been chatting to the players about just now.
"It would be easy to go in there and dismantle a lot of good things that have gone on at this football club and lay the law down, point fingers and tell home truths to make myself feel better.
"I know we need a reaction but results stand at me - I'll take that on board.
"When I came in I said when there are positives and accolades the players take them. I'll take the negatives and that is what I've got to do now. The players have to have the same belief to express themselves and play with a smile on their face."
- Sports & Recreation
- Daniel Nardiello