United made a disastrous start to their Boxing Day fixture at the KC Stadium as goals from James Chester and David Meyler put the Tigers 2-0 up inside the opening 13 minutes.
However, Chris Smalling reduced United's arrears and Wayne Rooney smashed in a great volley to equalise, before a Chester own goal sealed a 3-2 win.
Moyes was delighted with the spirit shown by his side, telling Sky Sports: "We didn't start well, but all credit to the players for getting themselves back.
"I think it's what Manchester United are capable of, they have been over the years.
"We played well enough after that to score goals, and create goals. It was a bit harem scarem in the last five minutes but I'd have taken that after being 2-0 down at the start of the game.
"The boys showed great character to come back."
It was a third straight league win for United and kept them in touch with the leading pack, but Moyes does not want to get carried away and has instead urged his men to keep working hard.
"We've got a long way to go. We can only try to hang on to the teams that are above us," he said.
"We're going to have to win an awful lot of games between now and the end of the season but the players are making a right good go of it and I believe we can be there or thereabouts at the end of the season and that is what we'll try to do."
Antonio Valencia was sent off in the closing stages after kicking the ball away to receive a second yellow card, but Moyes insists United can cope with his absence.
"It was something he shouldn't have done," said Moyes. "We'll have players to fill in, we've got a big squad so hopefully we'll all be ready to go again against Norwich on Saturday."
Winning when playing badly is the sign of a good team. Sometimes it's the sign of a bad team. You are sh1te.- popularpete