David Moyes' team was trounced 4-1 by City at the Etihad Stadium, with the striker scoring a late consolation after a Sergio Aguero double and strikes from Yaya Toure and Samir Nasri had put the game beyond doubt.
The result marked United's heaviest away defeat to City since 2004, and Rooney has acknowledged that his side's performance was far from acceptable.
"It is not good enough," Rooney told MUTV. "We know we have to improve and stop giving these sloppy goals away.
"It is not nice to lose to City. It was a bad result and we are all extremely disappointed."
Rooney's free kick was his fourth goal in three games and made him the all-time leading scorer in Manchester derby matches with 11 goals.
The England international was keen to focus solely on Wednesday's Capital One Cup clash with Liverpool, rather than personal milestones.
"It is nice to score but it means nothing," he added. "The most important thing today was the points and we have come away with none.
"Thankfully we have another massive game against Liverpool. It is the ideal game for us. Any game against Liverpool, whether it is a friendly, Capital One Cup or Premier League, is massive. We have to try and get the victory so we can put this defeat to the back of our minds."
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- Sergio Aguero
- Samir Nasri
- Yaya Toure