With United defending a 1-0 lead courtesy of Zlatan Ibrahimovic's first-half goal, Fellaini came on as a substitute in place of Henrikh Mkhitaryan in the 85th minute and conceded a penalty by fouling Idrissa Gueye, allowing Leighton Baines to equalise from the spot.
The costly error resulted in United playing out their third straight 1-1 draw in the Premier League to fall 13 points behind leaders Chelsea.
After the match, Mourinho defended his decision to bring on Fellaini to deal with the aerial threat posed by Everton, and Neville concurred that the move was tactically sound.
"It's just rubbish from Fellaini. It's obvious – as soon as you commit yourself in the box you're in danger of giving a penalty away. It's garbage from him," the former Manchester United defender was quoted as saying by the Sun.
"It's nothing to do with the manager. He's made the substitution for the right reason and the player has let him down. It is really pathetic from such an experienced player. He's come on to actually see a game out. No composure and it was obvious what was coming, you could see it a mile off.
"[His introduction] was purely to deal with the long balls into the box. You can see the sense in it but when he goes on and does such an idiotic thing like that. It was really poor from an experienced international. You can't blame the management, it's just a poor moment," he said.
Neville also spoke about United's poor run of form and declared that there was "softness" running through the core of his former club.
"Jose Mourinho will be losing faith in the sense they are playing quite well – although they didn't play brilliantly today – but not seeing it out. You have to see it out. That's the disappointment," he told Sky Sports.