Derby reaction: 'Rotherham United were better than us,' says Sheffield United boss
The Millers needed a stoppage-time goal to force a 2-2 Championship draw against high-flying Sheffield United at a packed AESSEAL New Stadium.
“Our performance was very, very poor but I need to compliment the opposition manager and their players," Wilder said. "If I was in their changing room, not getting the win, I would be disappointed.
“They won more first and second balls. We thought that maybe we could just stroll in here and it would be a pretty easy afternoon."
Warne pointed to the fact that visiting goalkeeper Dean Henderson was named Sky Man of the Match to show how close his team had come to victory in the televised clash.
"Their goalkeeper was Man of the Match which speaks volumes," he said.
“I think their dressing room would be happy with a point. At 2-1 down, it was a bit harsh on us. The lads don't give up. If it's only a one-goal deficit you're still in the game."
The Millers trailed to a Mark Duffy goal at half-time, Jon Taylor equalised in the 65th minute and Chris Basham headed the Blades in front after 85 minutes before substitute Jamie Proctor swept hom a 93rd-minute leveller in front of the kop.
Only the heroics of Henderson kept Wilder's team in the game.
"We ask the lads at half-time to keep going and give the fans something to be proud of," Warne said.
“I thought we started really well and they scored through our mistake (a poor Semi Ajayi pass), which was hugely disappointing. We were well rocked and the game swung towards Sheffield United.
“In the second half we were really good and on the front foot. We took the game to them.
“To get the equaliser late on made it feel like a win. We were always more than in the game. We caused them problems. If we had lost the game 2-1, I would still have been hugely proud of the performance."
Wilder added: “Apart from our goalkeeper, I could have made any changes. We changed the shape about three or four times. We could have pulled anyone off at any given time but we managed to find ourselves 2-1 up.
“It has not been a good day for us and I won't hide behind the result.
“The goalkeeper has kept us in it. He was absolutely outstanding.”