TOKYO—A construction worker at the crippled Fukushima Daiichi nuclear plant has cancer that could have been caused by radiation exposure, the government said Tuesday in announcing the first compensation award to be granted in such a case.

The man, who wasn’t identified, was diagnosed while in his late 30s with leukemia, the Health and Labor Ministry said. His current age and condition weren’t disclosed, but the ministry said he is receiving outpatient treatment.

The...