Edin Dzeko scored after just 43 seconds to throw the hosts' game-plan into disarray and the Red Devils were unable to offer a convincing fightback as they fell 15 points behind their local rivals, having played two games more.
"I think we've played a very good side," Moyes told Sky Sports after the game. "It's the standard we need to aspire to at this moment in time.
"I think we need to play better. We need to come up a couple of levels ourselves; at the moment we're not there.
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"In the opening part of the game we found ourselves under real pressure. They scored after 30 seconds or so, which made it an uphill task."
He added in his post-match press conference: "The key to it was not to lose a second goal to give ourselves a chance of getting in there. We didn't start the second half well, we brought pressure on ourselves. It was poor marking [for Dzeko's second] and we should have done much better.
"We never came out of the blocks. You have the players warmed up and ready but they never started well enough. I said after half-time we needed to play in their half of the pitch but we never really got that."
The Scot accepted "responsibility" for his side’s performance but warned that, while "disappointed" in their poor season, he will need plenty of time to whip the club back into shape.
“I take responsibility for the team and always will do," he continued. "We came into the game in pretty good form ourselves [but] conceding early put us on the back foot. It was nothing to do with lack of commitment or lack of effort.
"I'll take responsibility. I'm the one who plays them, picks them. There's a lot of really good players in the squad.
"Everybody knows this is going to be a job that will take a bit of time to get where we will like it. That's the job and I recognise that.
"It's disappointing. I thought it would be a tough year for us but I hoped it would be much more competitive and closer to the top than we are at this moment in time.
"The rebuilding is underway in its own way. A lot of other clubs have had to do rebuilding jobs; it's going to be a period of time to get to that level."