The Swiss justice ministry told AFP the Gambian man had been arrested at a centre for asylum seekers in the northeastern town of Kreuzlingen, not far from the Austrian border.
The man, who was not identified, "is suspected of having murdered a woman in Vienna," the ministry said in an email, adding that he was being held pending extradition to Austria.
The arrest comes after a woman, named as Lauren M. from Colorado, was found dead last week in her Vienna apartment lying face-down on a mattress with splatters of blood on the floor.
Her half-undressed body had been surrounded by a sea of burning candles, according to the APA news agency.
Swiss authorities said on Friday the suspect "will be questioned today by the prosecutor in the canton of Thurgau on behalf of the justice ministry," stressing that if he agreed to be sent to Austria, he could be extradited "immediately".
If he opposes the move, however, Swiss authorities will need a formal extradition request from Austria before determining if he can be sent out of the country, the ministry said.
The office of Austria's state prosecutor meanwhile said it had already filed a formal extradition request for the suspect on Friday, although Swiss authorities said they had not yet received it.
Lauren M. was first reported missing by her employers, who described her as "very reliable", after she had failed to pick up their child from kindergarten.
Firemen then went to the American's one-bedroom flat and broke down the entrance door, leading to the gruesome discovery.
Lauren M. had been living in Austria for several years and was a German language student.