Having trailed following Edin Dzeko's second-half strike, Aron Gunnarsson levelled from close-range before the 25-year old bagged a double with his head to secure the win.
The visitors were without captain Vincent Kompany in defence, but Campbell says that the Bluebirds did not underestimate the strength of their opponents
"They had a centre-midfielder playing centre-back so you know you're going to get a bit of opportunity there," he told Sky Sports 1HD.
"But Man City have got a great squad of players and they're all internationals so we knew it was going to be difficult, but we knew we'd get chances.
"There was some great deliveries into the box today and it was a great all-round team performance.
"It was very special (scoring twice) but the most special thing was getting the three points today. When you're a promoted side you don't want to wait too long to get points on the board so today was a massive bonus for us.
"The manager's been saying to us that we deserve to be in this league on merit. We did very well last season in the Championship and we've got to go out there and show everyone what we're about.
"We're going to be underdogs for a lot of the games this season but as we showed today we've got the determination and hopefully that will see us through."
Campbell's team-mate, Craig Bellamy, also hailed the performance which saw his side pull off a shock result.
He said: "It was a tough afternoon for us, Man City are such a good football team.
"We had to be patient, we knew we weren't going to have the ball for long periods.
"With Frazier today, not just his second-half goals, but in the first-half he created a lot of opportunities for himself. When you're a threat you always feel like you have a chance of maybe nicking a goal.
"It was a great goal by Dzeko, Man City are capable of that. As well as we defended, all it takes is that one great strike which he hit.
"But we kept our heads up. That's the thing about our players as well, we'll keep going no matter what."