Moyes was fired last week ago just 10 months into a six-year contract.
Fellaini worked with Moyes at Everton for five years and was the Scot's first signing when he took over at Old Trafford.
The Belgium international acknowledges the 51-year-old failed to deliver on the pitch, but that did not stop the midfielder from feeling sorry for his former manager when he was axed.
"It was a hard week for me (last week) because I was close with the manager," Fellaini told MUTV.
"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."
Ryan Giggs has been put in charge until the end of the season.
The Welshman has added fellow 'Class of '92' graduates Paul Scholes and Nicky Butt to his coaching staff.
Fellaini has been impressed with how the trio and Phil Neville, who stayed on as coach despite being initially brought in by Moyes, have taken to the task in hand.
"When they came in the first thing I saw was that the players respect the manager and I think that's important," the 26-year-old Belgian said.
"They were big players before and they know the club and the biggest manager (Sir Alex) Ferguson."
Fellaini did not make the bench for Giggs' first game in charge - a 4-0 win over Norwich - but hopes to be included in the Welshman's matchday squad for Saturday's home game against Sunderland.
"I respect the choice of the manager," the former Everton midfielder said when asked about being dropped last weekend.
"I've worked hard in the last few days to get into the squad, I did last week as well."