The report of the capture was last in a routine list of Taliban activities posted on the Web site.
"It is to be said that five days ago, a drunken American soldier who had come out of his garrison named Malakh, was captured by mujahedeen... He is still with mujahedeen," said the report.
The short message did not elaborate on his whereabouts or their plans for him, nor did it provide any proof of its claim.
The U.S. military earlier said it had intercepted communications in which insurgents talked about holding an American.
The soldier was noticed missing during a routine check of the unit on Tuesday and first was listed as "duty status whereabouts unknown."
His body armor and weapon were found on the base.
It was not until Thursday that officials said publicly that he was missing and described him as "believed captured." Details of such incidents are routinely held very tightly by the military as it works to retrieve a missing or captured soldier without giving away any information to captors.
Two U.S. defense sources said the soldier "just walked off" post with three Afghans after he finished working. They said they had no explanation for why he left the base.