Fellaini, who played under Moyes at Everton, said that it had been a hard week because he had been close with the manager.
According to the BBC, Fellaini, who had been the Scot's first signing when he took charge at United, said that he had been sad, but that is football as results are the most important thing in this sport and so the club had decided to make the choice.
The Belgium International, who had been left out of the United squad in the club's interim boss Ryan Giggs' first game in charge against Norwich City, said that they have a new manager now and are looking forward.
Fellaini said that United has three more games left in the season and they want to finish strong.
Giggs has been appointed as United's interim boss, following Moyes' sacking last month, and is being assisted by fellow 'Class of 92' graduates Phil Neville, Paul Scholes and Nicky Butt, the report added.