The former Southampton youth-team prospect, 23, joined the Bees in 2009 and went on to make more than 60 appearances for the London club, prompting an approach from the Bluebirds.
"We would have loved to have kept him but we wish him all the luck in the world," Rosler told the club's official website.
"He has the capability to step up to that level, he has a lot to learn but with Cardiff manager Malky Mackay and his coaching team, he is in good hands.
"We have all our young players on long-term contracts and if clubs come in for them, they have to pay.
"He had a good season last year and like Tom Adeyemi, who joined Birmingham City, he deserved the move upwards."
Moore moves to City for an undisclosed transfer fee while Brentford sporting director Mark Warburton has given Rosler the green light to find a suitable replacement.
- Sports & Recreation
- Malky Mackay