After falling behind to Edin Dzeko's opener, Aron Gunnarsson levelled proceedings on the hour before Campbell bagged a brace to the delight of the home fans in the Cardiff City Stadium.
Alvaro Negredo halved the deficit to set up a thrilling finale, but the Bluebirds held on for a famous victory - their first in the Premier League.
Campbell said: "The most special thing was getting the three points. When you're a promoted side, you don't want to wait too long to get any points on the board so today was a massive bonus for us.
"We knew it was going to be difficult but we knew we'd get our chances. (There were) some great deliveries into the box and it was a great all-round team performance.
"The manager's been saying to us we deserve to be in this league on our merit. We did very well last season (to get promoted from) the Championship and we've got to go out there and show everyone what we're about.
"We'll be the underdogs for a lot of the games this season but, as we showed, we've got great determination and hopefully that'll see us through."
Cardiff captain Craig Bellamy believes the crowd played a key part in Sunday's success against his former side.
"We can hold them responsible for the three points, they were incredible," he added on Sky Sports 1.
"This Premier League's so tough and not every week we're going to have Man City but one or two teams, we're going to need them to be as positive as that if we're going to try and stay in this league."
Manchester City boss Manuel Pellegrini was left frustrated at his side's inability to break down Cardiff's defence and hinted at his dissatisfaction with his own backline after Campbell scored from two corners.
"We played against a very difficult team. They were with 10 players in their own box and it was very difficult for us to try to score," said the Chilean.
"When we scored the first goal I thought we played a bit better after that.
"But their set-pieces decided the match. It was just two corners that decided the game."
Cardiff boss Malky Mackay was understandably delighted with his side's performance after watching them open their Premier League account.
"They were fantastic to a man, disciplined, did their job and deservedly won the game," he said.
"It's about being professional and not being caught up because we've got a fantastic crowd here. And for the first time in our history we're in the Premier League playing against a team who we go a goal down against and they only lost one game after going a goal down.
"They've got a tremendous clean-sheet record and we knew that it was going to be tough at that point, but we remained disciplined, organised and I felt there was a lot of belief in our ability today."
Mackay admitted he knew City's resolve would strengthen after falling 3-1 behind, but believes Cardiff proved they belong in the division after repelling last season's Premier League runners-up.
"I knew that they were never going to let their guard down and go to the last minute," he added.
"We were playing against top, top players. I've spoken to the boys about how proud they should be because they deserve to be here and they deserve to be out there playing against these top players."