It was the sixth, and most devastating home defeat of what has been a long and painful debut season for the United manager.
The last time United lost six league matches at home was in 1978 and that is a statistic which Rooney feels is unacceptable.
"It's not good enough," Rooney said in a frank interview with MUTV.
"We can't lose six homes games in a season and we have to put that right, and make this a place which teams fear again."
Edin Dzeko scored either side of half time before Yaya Toure added a third in stoppage-time to seal a win which will no doubt raise further questions about Moyes' future.
The home fans who were left inside Old Trafford by the final whistle did not boo their team off, but there was also a noticeable lack of songs in support of the United manager.
Rooney has pleaded with the United fans for patience despite what has been a dreadful season.
He added: "They were great again tonight. As a team, we need them to be strong and to understand there has been a big change at the club.
"We have to step up and start giving the fans something to cheer about with some good performances and wins."
United went into the match full of confidence after back-to-back wins against West Ham and Olympiacos.
But the wounds that were inflicted by Liverpool the previous week have now been reopened.
And while City push for the title, United's very slim hopes of a top-four finish must now been gone.
Rooney acknowledged United did not do enough to take anything from the game.
"It's a bad night and we didn't deserve to win," the United striker said.
"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.
"It's probably been the story of our season so far that we've got a couple of good results and then one which sets us back a bit."
United have a home game against Aston Villa on Saturday which is followed by a daunting double-header against Bayern Munich in the Champions League.
Rooney wants the team to bounce back quickly to avoid further embarrassments such as the one they endured on Tuesday night.
"We know we have to respond," the England striker said.
"It's a bad night for us all but we have to move on.
"We've got two big games coming up against Bayern Munich but first we have to focus on the Villa game and try to end the season on a positive note."
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