Phil Jagielka was delighted with his side's "fantastic" win and is thrilled to start the season in fine style, while Phil Neville also praised the Toffees goalscorer
The midfielder was excellent in a deserved win for the Toffees, scoring the only goal of the game in the second half with a fierce header, and his performance thoroughly impressed Jagielka.
"We knew it was going to be hard against a great team like Manchester United. Fortunately the big man [Fellaini] got a great goal and we had something to hold onto," he told Sky Sports.
"The team got a bit tired towards the end and it's great to send the Everton fans home with a win and not the doom and gloom we usually start with.
"It's nigh on impossible to mark him [Fellaini], in training we usually pull his hair, but obviously you can’t do that in a game!
"It’s a fantastic start for us, three points, as underdogs, we’ve got a big game coming up at the weekend, and we hope to get all three points at Aston Villa."
Fellaini was also pleased with his performance and felt Everton created plenty of chances against last season's Premier League runners-up.
"It’s a great start, Everton played well, started very well, kept going, it was a good performance," he added.
"It’s a good goal, a good cross, I think it’s good for me and good for Everton.
"Of course Everton had a lot of chances, first half we kept going, and I hoped the team would score."
Captain Phil Neville also added his thoughts, describing Fellaini as "unplayable" after the game.
"When he plays like that he is unplayable because of his sheer size. He puts himself about and the option he gives us it allows us to switch things up a bit," he said.
"With them having a lot of defenders out we were able to play a little more direct, mixed our game up a lot more and everything seemed to stick on top of him.
"When you have a centre-forward that can hold the ball it allows you to get up the field and when you are playing the top teams you have to get up the field and create chances, and tonight I think Fellaini was outstanding. He was probably the focal point of every attack we had.
"He's full of energy and he and [Nikica] Jelavic look like they could form a good partnership."