The game was settled by the hosts' skipper Johnny Mullins in the 57th minute as he rose above a crowded goalmouth to head home David Noble's right-wing corner.
"We could have been beaten a lot more," said Cook.
"We just didn't turn up. The best team won, comfortably by far. It was an embarrassment because of how bad we were.
"Players need to stand up now and again and at our club they need to stand up. Everybody who comes to play for us comes with a responsibility.
"You can dress it up as a manager but the goal was coming. They were going to score and what we didn't want to do was wait for them to score to make changes. But whatever we were trying it wasn't good enough.
"Every credit to Rotherham. They were a handful from the start and it was pure luck that we went in at half-time on level terms.
"Unfortunately for us we let our supporters down without a shadow of a doubt, but I hurt more than our supporters.
"You have got to have desire, you have got to have pride, you can't turn up at these places and expect it to happen, but I carry the can for us because I am the leader of this group and we weren't good enough."