The reigning champions are seventh, 18 points adrift of leaders Chelsea after City inflicted United's sixth home league loss of the season.
"I am the one who picks the team," Moyes said. "I take responsibility and always will do. It is disappointing.
"I thought it would be a tough year but I hoped it would be more competitive."
The defeat means United have now lost more league games at home this season than in the previous three campaigns combined.
In nine matches against the Premier League's top five, they have won just once, against Arsenal in November.
Despite reaching the quarter-finals of the Champions League, United have fallen well short of the consistency which saw them win the title by 11 points under Sir Alex Ferguson before his retirement last summer.
"Everybody knows this is a job that is going to take a little bit of time to get the way we would like but that is the job," added Moyes, 50
.
"Other clubs have had to do rebuilding jobs - we hope it won't take as long as some others but we have a period of time to get to that level.
"There are a lot of really good players in the squad, a lot of internationals, who, on their day, can be a match for most players."
The former Everton boss admits he hoped United would be "closer to the top of the league than we are at the moment," but believes his side did not "lack effort or commitment" against their city rivals.
City, who were the last team to deny United the title - in 2012 - scored through Edin Dzeko after 43 seconds at Old Trafford and the Bosnian doubled the lead before Yaya Toure struck a third late on.
"We never came out of the blocks," added Moyes. "We never got started. For Manchester City to score inside the first minute gave them a really big lift.
"I told the players at half-time that we wanted to start the second half stronger and that we needed to play in their half but we never really got that."
United striker Wayne Rooney said the team's record at Old Trafford is "not good enough", adding: "We can't lose six home games in a season. We have to put that right and make this a place teams fear again.
"We know we have to respond. It's a bad night for us all but we have to move on."
Speaking to MUTV, Rooney added: "We didn't deserve to win. We weren't clinical enough and we didn't create enough chances.
"No-one likes to lose a derby game - they're big games and they're great to win, but when you lose it's not a great feeling."
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