The 18-year-old, who spent a period on trial with the Championship club last season, has overcome work permit issues to sign a deal until 2015.
Goalkeeping coach Billy Mercer told the Burnley's official website: "Daniel has been on our radar but his parents have done a great job, along with [Burnley CEO] Lee Hoos, in getting his visa sorted and he is here for the long haul as one for the future.
"It has been a big turnaround for us this summer, with Lee [Grant] going off to Derby, Brian [Jensen] leaving after 10 years and things not working out for Jon [Stewart].
"With experienced goalkeepers leaving, it was important that we got replacements, not only for the next couple of years but hopefully beyond that."
Nizic had also been linked with a move to Portsmouth.