City thumped the Millers 7-0 to move into the fourth round.
Speaking afterwards, Warne was quick to single out the contribution of youngster Ben Wiles who belied his tender years, by continuing to make his mark on the game despite coming up against the likes of Kevin De Bruyne in City's midfield.
“I thought Ben Wiles was great in midfield. It was just his third game. He did not show them too much respect. I'm really pleased with him.
“We are not embarrassed by the score line. I spoke to Pep (Guardiola) before and after the game and I shook his hand and appreciated the respect he showed our team by putting a strong team out.
Following the final whistle, Warne took time out to acclaim the large Millers travelling contingent, who made the trip over the Pennines and continued to support their side in trying circumstances - which was a point not lost on the Rotherham chief.
“The support was hugely impressive," Warne enthused.
“I felt I let them down a little bit as they did not have a lot to shout about. I wish we could have got a goal. They stayed with the team. They appreciated what our lads tried to do against world-class players."