United defeated City 2-0 to mark Carrick's first game in charge after he was named manager to the end of the season last week.
"I wasn't at the stadium, I was in the BBC studios, but did you see the difference in the stadium? Even when he walked out before the game and he's trying to pump the crowd up," Rooney said on the latest edition of BBC podcast The Wayne Rooney Show.
"I think the fans felt something yesterday. They got a taste of what it was like under Sir Alex Ferguson. Some of the young fans who have been through a terrible time over the last 10, 13 years, really, since Fergie left.
"They haven't witnessed a team perform that well. I've been part of some of those teams as well. I've played in some of them.
"They haven't witnessed a team perform that well, with that energy and that belief in their fighting and being brave with the ball, but also being resilient and tough to play against.
"Them fans felt that yesterday and hopefully that's a glimpse of what we've got coming in the next 16 games."
Rooney added: "You can feel it from the fans as well, the difference in energy from the team, the confidence, the belief, the composure on the ball at the right times.
"It was night and day from what we've been seeing, so it was such a good day. It reminded me, when we talk about Man United's DNA, that was it...
"Work hard, run for your team-mate, get back into shape, be hard to play against - when you get an opportunity run forward."