Moyes made a slow start to life as Sir Alex Ferguson's successor at United, winning just two of his opening six Premier League matches at the club.
However, the Red Devils have now won four of their last five league games and are currently on a nine-match unbeaten streak, a run that includes Sunday's 1-0 victory over league leaders Arsenal.
"There was a little bit of criticism going on but I am pleased I didn't have to answer it because David has done it himself," Charlton told United Review.
"We are not the type of team who can languish in the bottom half of the league and it is promising to see us putting together some good results. We have had a lot of publicity from changing manager after so long and I've had the sense that teams have maybe been upping their game a little bit against us.
"But we have been first-class recently. Here we are, creeping up the table again and that is what we want."
United now sit fifth in the Premier League table, five points behind Arsenal.