But Curbishley was vindicated two minutes later when Cole fired the winner from a cross by Freddie Ljungberg. Bobby Zamora had given West Ham a 20th-minute lead with a close-range header but Tyrone Mears levelled for the rock-bottom Rams in the 64th minute.
The criticism stung clearly stung Curbishley who called on the club's fans to show more realism over a season that has seen his squad ravaged by injuries and illness.
"We are delighted to get the result," said Curbishley. "But I am a bit disappointed because there has not been too much appreciation of what has gone and what we have had to put up with this year.
"I don't think there is much appreciation going on at the moment and I can't do too much about that. The players are very disappointed with it. But I know what happens here. You put the results together and it turns the other way.
"When the team is winning, people don't like coming here because of the backing we get. When they are not happy they let you know. But surely there has got to be some realism and appreciation of what has happened to us this year - week-in, week-out we've been under the cosh. I have been very appreciative of the players who have been playing because we have had loads of problems. We've had players missing all year and I think we had 14 out today.
"If it is not good enough for some people then we have just got to get on with it, I can't do anything about it. That sort of chant is common place up and down the country and it is not the best of chants if you are the manager, I must admit.
"We've just got to get on with it and it can turn around as quickly as it happens. I am disappointed with the reaction and the lack of appreciation. The season has been one of frustration for everyone. It has been a catalogue of disasters. I felt we needed a bit more up front and I am pleased that Cole came on and got the goal.
"But I have been around long enough to realise if they are not happy here they will let you know."
Derby boss Paul Jewell was delighted with their second-half performance but he knows his biggest job will be undertaken in the summer.
Derby are already preparing for life in the Championship and Jewell is adamant that the mentality of his team will be changed during the break.
"I thought we played well in the second half," said Jewell. "I think West Ham were hanging on at the end.
"We just didn't have that final pass or killer instinct but the second half performance augurs well for next season."