"It's the worst feeling in football," Allen told The Liverpool Echo. "These are the kind of games you have to be on it mentally for the whole game but we started slowly.
The Reds' defeat at Boundary Park was they first time they have lost an FA Cup tie to a team from outside the top two tiers since Bolton won a third-round replay at Anfield in January 1993.
The former Swansea man has dismissed claims that the changes made by Brendan Rodgers for the game could be used as an excuse for their performance and the result.
"To concede three goals in the manner we did was bitterly disappointing," Allen told the newspaper. "We still had a strong side out. The manager made changes but for those of us who came in it was a chance to impress and stake a claim. A lot of people will look at themselves and be disappointed that they didn't take that chance as well as they could have.
He added, "I don't think we underestimated them. But, no disrespect to Oldham, when you've been knocked out by a team in League One, then something has definitely gone wrong somewhere.
"It hurts and I feel sorry for the fans. Once again, they turned up in their thousands and they have gone home very disappointed."