It has been a testing period for the former Everton manager since he replaced Alex Ferguson at Old Trafford, with United making their worst start to a Premier League season.
However, they are unbeaten in all competitions since a 2-1 home defeat by West Bromwich Albion on September 28 and have climbed to sixth in the Premier League, one point off the top four.
Conceding a late equaliser in a 2-2 draw at Cardiff City last week was a jolt, but United responded with a 5-0 thrashing away to Bayer Leverkusen on Wednesday -- a result that could prove to be a defining moment in the season.
United visit Tottenham Hotspur on Sunday when the London club will be anxious to bounce back from the 6-0 mauling they suffered at Manchester City last week.
A relaxed-looking Moyes said he felt there was still room for improvement from his United side.
"The best is to come and I still think we can continue to improve and progress," he said.
"We had some little bits of inconsistency at times, which we need to try to do better with, but there have been moments that have shown we are beginning to come right back into form.
"That is what I hope. Midweek certainly showed what we are capable of when we are playing well and the job is to do that week in, week out."
Experienced left back Patrice Evra said United were determined to use the Leverkusen result as a springboard.
"We were so disappointed about the Cardiff draw. We are in good form but for that silly goal at Cardiff," he said.
"We want to keep winning and winning. We cannot wait for Tottenham."
Evra said that the champions would again be pushing for the Premier League title despite their slow start.
"People always write Manchester United off," he said. "Remember last year people were saying the same then, but in the end we won the league by 11 points."
(Reporting by Martyn Herman, editing by Mitch Phillips)
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