Mourinho was speaking after his side’s 4-1 win over Burton Albion on Wednesday, where Marcus Rashford (5, 17), Jesse Lingard (36) and Anthony Martial (60) were all on target.
The club were even victorious in last season’s competition, as United defeated Southampton.
But speaking to the club’s official website, Mourinho said: “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.”
Up next for United is a fourth round clash with Swansea City.
Mourinho added: “We did our job. Not a phenomenal performance but a solid performance. Since the beginning, we played in a serious way. The players, many of them, needed 90 minutes to play because many of them were not playing a lot during the season. I think we did our job professionally.
“What pleased me the most was the commitment. They took it in a serious way. While the result was open, until we were 2-0 or 3-0 up, they forced the game, they wanted to kill the game. They wanted to be in the next draw. That professionalism and commitment is a very important thing.”