Both the Red Devils boss and Burton Albion counterpart Nigel Clough made nine changes to their sides ahead of their third-round encounter at Old Trafford on Wednesday.
Following the game, Mourinho told BBC Sport: "If the competition is an official competition then it is important for Manchester United and for me as a manager.
"If you ask me could the English football survive or even be better without this competition? Maybe, maybe we would be fresher for European competition for example.
"But we have this competition, we have to respect the sponsors, we have to respect the opponents and a lot of us are trying to do our best."
United will face Swansea City away in the fourth round of the competition, with the tie to be played in the week commencing October 23.