The Wearside club have not won a trophy since their 1973 FA Cup success and O'Neill fielded a full-strength side in order to maximize Sunderland's chances this season.
However, Marvin Sordell scored twice as the Championship outfit shocked the Black Cats in their third-round replay at the Stadium of Light, and O'Neill has admitted that he is annoyed by their exit from the competition.
"This is a big club and we shouldn't be spending 40 years without winning something, we really shouldn't," said the Sunderland manager.
"We didn't take anybody lightly here tonight, we just got beaten because we didn't create enough chances in the second half.
"I know some people field weakened sides but we didn't, we tried to win tonight and we weren't able to do it, and that was very disappointing."
Bolton will host Everton in round four.