While City's 3-0 success on Tuesday took them to within touching distance of the Premier League summit, United's defeat left them 12 points below the Champions League places with only seven games of the season remaining.
United have fallen a long way since romping to last season's league title by 11 points, but Moyes rejected suggestions that the rebuilding process at Old Trafford would take longer than anticipated.
"It is under way in its own way," he told a post-match press conference.
"You don't just suddenly change things around
As I said the other day, a lot of other clubs have had to change and they have had to do rebuilding jobs and look at the time it has taken them to do that or get to a level of competing.
"We hope it won't take us as long as some of those clubs have taken
I think we have got a period of time where we are going to have to make sure we get to that level, which we are not at just now.
"Everyone knows this is going to be a job which is going to take a little bit of time to get the way we would like it, but that is the job and I recognise that."
City took the lead after just 43 seconds via Edin Dzeko, who added a second goal early in the second half before Yaya Toure sealed the visitors' third successive win at Old Trafford in the 90th minute.
United had gone into the game on the back of morale-boosting wins over Olympiakos and West Ham United, and Moyes was at a loss to account for his team's limp start to the match.
"I just think we never came out of the blocks," said the Scot, whose side host newly recrowned German champions Bayern Munich in the Champions League quarter-finals next week.
"You prepare the players, you warm them up, you do all the things to have them ready, but we just never started
It gave them (City) a real big lift to get a goal so early on."
- 'I take responsibility' -
United director Bobby Charlton confirmed the board's backing for Moyes in an interview with the BBC published earlier on Tuesday, declaring that he was "absolutely certain" that the former Everton manager was "the right man" for the job.
For his part, Moyes was prepared to accept the blame for his side's latest setback and dismissed the notion that the title-winning squad that he inherited from Alex Ferguson was not up to scratch.
"I take responsibility
I have to be the one who plays them, picks them, and that is what it is," he said.
"I think there are a lot of really good players there."
City now trail Chelsea by only three points and have two games in hand -- at home to Aston Villa and Sunderland -- but manager Manuel Pellegrini said it was inaccurate to suggest that the title was theirs to lose.
"I don't think so," said the Chilean, whose side visit fourth-place Arsenal on Saturday.
"The title race continues
We continue fighting with all the other three -- Arsenal, Chelsea, Liverpool
We have two games postponed, but we have to win those two games.
"Tomorrow (Wednesday) I start thinking about Arsenal and then I think about the end of the season."
United midfielder Marouane Fellaini appeared fortunate to escape with only a booking after catching Pablo Zabaleta with an elbow in the first half, but Moyes claimed not to have properly seen the incident and Pellegrini said he had no complaints with referee Michael Oliver's decision.
"I didn't see, but it is the referee that has to see those things, not me," he said
"If he said it is not a sending-off, I respect that."
Source : AFP
Source: AFP