Television pictures showed that Dragovic, standing behind Federal Council (cabinet) member Ueli Maurer, gently slapped the balding politician on the back of the head as he was handing out medals to other Basel players.
When Maurer, who is also responsible for civil protection and sport, turned around to see what happened, Dragovic quickly turned his head in the other direction amid laughter from the other Basel players.
Apparently oblivious to what had happened, Maurer then continued handing out medals.
Basel said the 21-year-old Austrian international had been fined for the incident and the money donated to charity.
"He realised in retrospect that his gestures were utter nonsense and showed a lack of respect for Federal Councillor Ueli Maurer," the Swiss club said.
"Basel joins in with this apology and reiterates that it does not tolerate in any way the behaviour of the player in this incident."
Basel beat FC Luzern on penalties after a 1-1 draw to complete a league and cup double.