The Belgian box-to-box midfielder scored the winning goal by rising highest to thump home a corner in the 57th minute.
Speaking to Sky Sports News afterwards, Moyes said: "I thought the players did really well throughout the night. We had a lot of chances and their goalkeeper (David de Gea) pulled off some good saves.
"[Fellaini] was excellent tonight but we know he's capable of that. Don't forget Manchester United were missing a lot of central defenders this evening.
"It was all about Manchester United but let's not forget that Everton are a good side as well and Goodison [Park] is a difficult place to come."
The Toffees will look to continue their positive start to the new season when they travel to Aston Villa on Saturday.