The 32-year-old from Seoul was issued a ban retroactive to August 31 so An will be able to return to competition on December 1.
An tested positive, the tour said, for a substance banned by the World Anti-Doping Agency (WADA) that was contained in the cough medication.
"He has cooperated fully with the tour throughout the process and accepted his suspension," the tour said in a statement.
An will miss autumn events that are mainly targeted at players trying to earn their tour status for next season, something An has already done.
An played in 31 events during the 2022-23 season and matched the best PGA finish of his career by sharing second last August at the Wyndham Championship. He also shared third last July at the Scottish Open.
World number 53 An won his biggest title at the 2015 BMW Championship on the European Tour.
In 2009, An became the youngest winner of the US Amateur at age 17.
