A double from Edin Dzeko and Yaya Toure's late strike sealed a comfortable victory for Manuel Pellegrini’s men at Old Trafford, which has put them three points behind Premier League leaders Chelsea with two games in hand.
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The loss was United’s 14th of the season, while pressure continues to grow on manager David Moyes following the club’s inconsistent results.
"[The fans] were great again tonight," Rooney told MUTV.
"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.
"It's a bad night and 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. For the fans and for us, we have to win more and stay together."
When asked about the cause of United’s sub-par campaign, Rooney replied: "I don't know but it's not good enough.
"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.
"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. We know we have to respond. It's a bad night for us all but we have to move on."