Christophe Berra, Daryl Murphy and Aaron Cresswell put the visitors in dreamland inside the first quarter of an hour and, even when Zak Whitbread reduced the deficit, Murphy added his second of the night to put Town firmly on course for their first away win of the season.
New Rams boss Steve McClaren was not due to officially begin his role until Wednesday - with academy manager Darren Wassall in charge at Pride Park - but he took it upon himself to go down to the dressing room at the interval and deliver a rallying call.
McCarthy said: "That hurts, and if I'm being honest it feels like a defeat.
"I think that's what they call a game of two halves, isn't it? The first half was absolutely fantastic for us and we deserved to be in the lead, absolutely.
"But in the second half we made mistakes and the goal at the start of the second half gave them hope - they had nothing to lose.
"Then they attacked us and attacked us and in the end what should have been a win has turned out to be a hard-fought draw.
"We took our foot off the gas. We didn't start the second half well, we didn't win a couple of tackles and then lost a couple of headers and our shape wasn't right.
"We made mistakes and let them in. We were bad at the back when we've been particularly good there ever since I came through the door.
"We've dropped two points, no doubt about that, and we are going to have to pick them up from somewhere else.
"That all sounds negative but there are lots and lots of positives from that performance too in the first half.
"There were things that were really, really good and there were things that were really, really bad and I'm sure they will be saying the same thing about their performance.
"But they will be considerably happier than we are, although in the end I think they deserved a draw for their second-half performance."