Marcus Rashford's brace was complemented by goals from Jesse Lingard and Martial on a night when Lloyd Dyer netted a late consolation in a win as comfortable for Mourinho's Reds as the scoreline suggests.
United's reward for beating Burton is a trip to Swansea in the fourth round, with Mourinho happy to come through the tie without any injury concerns or cards.
The manager was quick to praise his attacking quartet of Rashford, Juan Mata, Lingard and Martial, with the latter looking sharp throughout the 90 minutes.
"He is starting lots of times," Mourinho said of Martial. "He started in the Champions League a few days ago, so he is always playing.
"I think he played every match, some starting, some from the bench.
"Of course I am happy with his performance, with his happiness on the pitch.
"He wanted to do things - a few things he did amazingly well, others not so well but he was always trying and, for me, that commitment, that happiness on the pitch is very important."