The Belgian rose highest to power home a header from Darron Gibson's corner 12 minutes into the second half at Goodison Park.
Speaking to Sky Sports News afterwards, Fellaini said: "It's a great start [to the season], we played well. We started well and kept going and going.
"We knew it was going to be hard against a great team like Manchester United."
The midfielder didn't take long to tweet a message as well, posting: "Hey Guys... First championship game, we were on the stage... Everton 1-0 Manu game over thx a lot... "
The Toffees continue their campaign at Aston Villa on Saturday.