The reigning champions made an unspectacular start to life under the former Everton boss, winning just two of their opening six Premier League games and falling to 12th place in the table.
But, after winning four of their last five league outings, including victory over table-toppers Arsenal, Neville says the 50-year-old is stepping out of the shadow of predecessor Sir Alex Ferguson and gaining the trust of the players.
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"Now the players are beginning to trust and believe in what the manager is trying to do. He is trying to make them better.
"It was always going to take a bit of time after one man had controlled the club for so long, there was always going to be a transition period."
After initially having been wary of making too many changes to a title-winning squad, Neville believes Moyes is now slowly starting to mould the United side into his own image.
"When you go into the club where they have won the league so many times and they have got so many great players, you don't try and change too much too soon," he added.
"There's not much that needed changing but the boss wanted to stamp his own authority on the team and I think against Arsenal, we saw that. I thought we saw a team that mirrored David Moyes."