Robertson has penned a four-year contract with the Championship club.
The midfielder is the nephew of former Cardiff player Gavin Rae and said: “He (Rae) told me a lot of good things about the city and the club.
"Going forward in my career now, this feels like the right place to be. I'm buzzing. It's been a few weeks coming. I'm so glad it is over the line."
Cardiff boss Erol Bulut added: "We're delighted Alex is here.
"There were six other clubs that were in for him, which shows what a talent he is. We had a meeting 10 days ago and explained our goals for the club and for Alex. Our hopes and plans were aligned.
"Alex can deliver a lot. He's got quality. He’s technically very good with a great work rate and can add to what we already have. We now have a great mix of youth and experience in the midfield."