United came into the game on the back of two successive defeats to start the season.
They lost 2-1 to Brighton and Hove Albion at home on the opening weekend and followed that up with a spiritless 4-0 loss at Brentford.
Buoyed by the presence of new signing Casemiro from Real Madrid in the Old Trafford crowd, the Red Devils put up a performance full of energy as goals from Jadon Sancho and Marcus Rashford gave them their first win of the season.
A Mohamed Salah header for Liverpool late on ensured a nervy finish but Manchester United held on.
"I'm absolutely stunned, to be honest with you," Neville said in his role as a Sky Sports pundit.
"I didn't think that was possible, tonight, what I saw, in terms of response.
"I thought the players had cemented what they do now over many months at the end of last season, the start of this season.
"What's happened this week I don't know, but they've responded. Maybe the slight changes in personnel, making brave decisions...
"The captain [Harry Maguire], the star player [Cristiano Ronaldo] have both been left on the bench. That means the manager just gains a little bit of gravitas, maybe, with other players.
"Bruno [Fernandes] has responded; I think [Lisandro] Martinez at the back has played well alongside [Raphael] Varane. I don't know what it is, but I can't believe what I've just witnessed.
"Liverpool weren't at their best, let's be clear, but they weren't at their best against Crystal Palace and Fulham.
"It was just the effort difference that we saw from Manchester United; they've outrun Liverpool, they've outsprinted Liverpool.
"We've questioned that a number of times over the last 12 months, that they're not capable of doing it, and they've shown tonight that they are capable of doing it."
