Moyes lasted just 10 months into a six-year contract after United decided to pull the pin last week following a disastrous season that has left the club languishing in seventh position in the Premier League.
Fellaini, who has also come in for heavy criticism this season for a number of unconvincing performances, was Moyes' first signing at Old Trafford following his arrival from Everton for a reported fee worth £27.5million.
"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."
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 sad 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."
