The Bluebirds shelled out a club-record £8.5m transfer fee for the 20-year-old, who Mackay claims had attracted the attention of three English sides.
"Cornelius has great potential and is a very good capture for the club," Mackay told reporters. "We tracked him all last year. But by Christmas, he was in such good form we really didn't think we had much chance because top clubs in Europe were watching.
"We were in competition internationally with another club and then three Premier League clubs moved in to try and take him as well.
"But we were able to do things quickly and I have to give a special mention to Copenhagen. They were tremendous - efficient, professional and friendly."
Cornelius has been capped seven times at senior international level for Denmark, scoring one goal in the process.