United have had a mixed start to their first season under Moyes and slumped to 12th in September their crushing 4-1 defeat at Manchester City.
But they are unbeaten in nine games in all competitions and defeated league leaders Arsenal 1-0 in their last fixture before the international break.
Moyes says his faith in the side was never broken and he is confident they can continue improving as the season progresses.
"I never thought we were out of it (the title race)," said Moyes, who takes United to Cardiff City on Sunday afternoon.
"We just go about our job and try and win each game. Everybody knew at the start what it was going to be like.
"We knew we had a change of manager and there were always going to be problems, no matter what - whether it be winning games or different personnel in the team.
"I have said many times that Manchester United have proved over the years they improve as they go on.
"My history at Everton was as a slow-starter as well. Hopefully we're beginning to find our feet and we'll continue to improve."