In a brief despatch, the agency said the American, identified as Miller Matthew Todd, 24, had been taken into custody on April 10 for "his rash behaviour in the course of going through formalities for entry" into North Korea.
The news agency added that he remained in detention and was under investigation.
The two-week delay in announcing the incident suggested it was timed to coincide with Obama's visit.
Bae was arrested in November 2012 and sentenced to 15 years' hard labour on charges of seeking to topple the government.
According to KCNA, Miller had a tourist visa, but tore it to pieces and shouted that "he would seek asylum" and had come to North Korea "after choosing it as a shelter.