Goals from Tim Cahill and Mikel Arteta ensured a first league win over the Reds since 2006 and Neville believes Everton must be aiming to make victories over their near neighbours a far more regular occurrence.
"I came to the club five years ago and the gap between the clubs was big. In terms of performances, we were fighting to compete," Neville told evertonfc.com.
"The quality gap was big, and the manager wanted to close that more and more. I think it has been happening. Last season we felt we were closing on them, and after yesterday we feel it is closing again.
"Even though we're rivals and Liverpool are the team our fans love to hate, they're still a team we want to get above. We can't disrespect Liverpool by saying they're gone or they're finished, because we could have still been in the bottom three without this result," added the Everton skipper.