Moyes endured a tough start to life at Old Trafford and was beginning to feel the heat when back-to-back losses against Newcastle and Everton saw them slip well off the title pace.
But things have picked up throughout December and United have now won five on the trot in all competitions, including Boxing Day's 3-2 success at the KC Stadium.
But United hit back, inspired by an all-action display from Wayne Rooney, who set up Chris Smalling's goal, equalised with a stunning volley that brought up his 150th Premier League goal for the club and pressured Chester into the own goal that sealed the result.
To collect three points from such a difficult start required the kind of self-belief and determination Moyes prides himself on and he goes into Saturday's clash at Norwich confident the club are finally finding their identity.
"I never doubted us," he said.
"Hopefully my teams have always been committed to trying to win games of football.
"After being 2-0 down, to get back to 2-2 was terrific but I actually felt a bit disappointed not be winning 3-2 by half-time because we had a few chances to do something more."
Moyes still has some loose ends to tie up from the trip to East Yorkshire and must find a new right-back against Norwich.
Rafael sustained a groin injury in the first half against Hull and is unlikely to be fit after just a day's rest, while Antonio Valencia, who dropped back from midfield to cover the position, is suspended after being sent off in injury time.
The likeliest solution involves switching Smalling from centre-half and bringing Nemanja Vidic back into the starting XI, but Moyes is not worrying about the reshuffle.
"I'll find a right-back, we have a big squad at Manchester United," said the Scot.
"We've plenty of players and plenty who can play on Saturday."
The former Everton boss is already enjoying his best run of results as United boss and a sixth successive victory at Carrow Road would give the distinct appearance of a team on a roll.
Moyes, though, was playing down such talk.
"I think all the games at this time of year are difficult to call," he said. "I expect Norwich to be another hard game.
"I think at this club you're waiting on winning more games than this and then you might talk about being on a run.
"We'll just try and win the next one and see what happens.
"All we can do is keep knocking away and hopefully be in there fighting come the end of the season."
Smalling is approaching the Norwich match in confident mood.
He said: "I think (Hull) is going to be a pivotal three points and it keeps us on this run as well, which is important because it is difficult when you have to play again two days later.
"It's always easier when you go into a game off the back of a win.
"We had belief going into this game, and after this win we'll continue to pick it up. I think we've had our setbacks this season, we need to make sure that we don't have any more and that we keep winning."
- Sports & Recreation
- Manchester United
- David Moyes
- Chris Smalling
- James Chester
- David Meyler