Barkley made his Premier League debut and was the Toffees' best player as they slipped to a 1-0 defeat at home to newcomers QPR on Saturday.
the 17-year-old put many of his more senior team-mates to shame with his endeavour and enthusiasm, and Neville said the youngster should have got more help on his maiden appearance.
"Ross was outstanding without going too much overboard," said the 34-year-old.
"People say bravery is going around kicking and fighting people, but it is taking the ball when you are 1-0 down and showing you can play football and he did that.
"I think everyone is quietly confident we have got something really special.
"This club produce not only good footballers but really good down-to-earth lads and he wants to learn and if he keeps his feet on the ground he has a really good chance.
"Ross is lucky in that he is going to get his chance because that is what the manager does, he gives youth a chance.
"It is up to us to support him because sometimes young players come on and do take the fight for the team - I have been there myself - and it is something you enjoy.
"At the moment you just say let him play and be free and there will be dips but we have to make sure as experienced players we are there supporting him.
"But he can be well pleased with his debut."
Barkley got his opportunity because Marouane Fellaini and Mikel Arteta were only deemed fit enough for the bench.
He started on the left of midfield before switching to the right in the second half but wherever he was playing he was not overawed by the occasion.
It was his run cutting in from the left and twice feigning to shoot which won the free-kick from which Leighton Baines rattled the crossbar.
It was the closest the hosts came to scoring all afternoon, although Tim Cahill and Jermaine Beckford should both have done better with headers late in the first half.