Moyes was sacked just 10 months into a six-year contract following a disastrous season that has left the club languishing in seventh position in the Premier League.
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Having spent five years together at Goodison Park, the Belgium international admitted it was disappointing that his working relationship with Moyes had come to an end.
However, Fellaini conceded United officials had little option but to axe the Scottish manager on the back of their worst campaign in recent memory, which also means they will miss the Champions League for the first time in 19 years.
"It was a hard week for me (last week) because I was close with the manager," Fellaini told MUTV ahead of Saturday's clash against Sunderland.
"I was sad, but that's football. Results are the most important thing in football so the club decided to make the choice.
"Now we have a new manager and we are looking forward.
"We have three more games left in the season and we want to finish strong."
Outgoing Netherlands coach Louis van Gaal is reportedly set to take the reins following the World Cup but, in the meantime, United legend Ryan Giggs has been named as interim manager until the end of the season.
Working alongside fellow club legends Paul Scholes and Nicky Butt, Giggs' first match in charge resulted in a thumping 4-0 win over relegation candidates Norwich City and Fellaini insists the successful trio had the respect of the United players from day one.
"When they came in the first thing I saw was that the players respect the manager and I think that's important," said Fellaini.
"They were big players before and they know the club and the biggest manager (Sir Alex) Ferguson."