Moyes, who was fired last week just 10 months into a six-year contract, made Fellaini his first signing at Old Trafford, paying Everton £28m for his services.
"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."
Belgium international Fellaini did not make the bench for Giggs' first game in charge - a 4-0 win over Norwich - but hopes to be involved against Sunderland this weekend.
"I respect the choice of the manager," he said when asked about being dropped for the visit of the Canaries to Old Trafford.
"I've worked hard in the last few days to get into the squad, I did last week as well. The players respect the manager and I think that's important."