The Premier League title challengers drove at the reigning champions from kick-off and were rewarded by an Edin Dzeko opener within the first minute - the quickest goal United have ever conceded at home in the Premier League era.
Moyes' men slowly regained their composure but it was a similar story after the break as Dzeko doubled his tally with United labouring 11 minutes into the second period.
Yaya Toure thudded home his 21st goal of the season in the last minute but Moyes was more concerned with events that took place in the first half.
"We never started well and never gave ourselves a great opportunity to get into the game," he said.
"Man City started really fast and with us conceding a goal after 30 or 40 seconds it made it difficult.
"We had to just try and make sure we got through that 10- or 15-minute period when they were out of the blocks.
"I thought after that we weathered it and got ourselves back into the game, finished the half quite strong."
Moyes' frustrations increased as Dzeko volleyed in from a Samir Nasri corner delivered in almost identical fashion to one that saw Fernandinho head over moments earlier. On both occasions, United's marking left a lot to be desired.
"The key to it was not to lose a second goal so we could always give ourselves a chance of getting in there and, again, I didn't think we started the second half well," added Moyes.
"I thought we brought pressure on us by our play and, in the end, I think we conceded a corner just before that (the second goal) - there was another one and it led to the goal.
"It was obviously poor marking and we should have done much better."
Reflecting further on a 10th league defeat this term, Moyes added: "We just never came out of the blocks - you prepare the players, you warm them up, you do all of the things to have them ready when they go out.
"It's something you've got to be aware of, they've got to ready, and we'll try and achieve that the next to time to make sure it doesn’t happen again."
