The U's, who have now lost six of their last seven games went behind in the 33rd-minute when Jean-Yves M'Voto headed powerfully in off the woodwork from Lee Croft's corner.
James Wesolowski drilled wide before Oldham went further ahead just after the hour when Derbyshire's shot was saved by Cousins and the ball bobbled in off U's defender Brian Wilson, after Croft's pass had dissected the U's defence.
Dunne said: "We're not winning enough battles at the moment as individuals. It wasn't good enough from us and it was very disappointing, especially as we had two weeks without a game prior to it.
"We didn't pass the ball well enough and once they got a foothold in the game, it was difficult for us.
"We changed it at half-time and went to 4-4-2 and I asked them to be brave, but I don't think that worked, either.
"We conceded a freak goal and it's probably one that's going to be shown over and over again and we're going to be laughed at.
"The keeper made a good save, but there were slow reactions and it was like it went in in slow motion.
"Nobody seemed to take responsibility to get to it and it's in the back of the net and I thought that generally the back four were poor."