"He was handed over to the Austrian authorities this morning at the border," Swiss justice ministry spokesman Folco Galli told AFP.
Swiss police had arrested the man, whose identity has not been released, on February 4 at a centre for asylum seekers in the northeastern town of Kreuzlingen, not far from the Austrian border.
Authorities decided on March 9 that he could be extradited to Austria, and he had not opposed the move, Galli said.
The man is suspected of killing a woman from Colorado who had been living in Austria for several years, studying German and working as an au pair.
Her half-undressed body was found on January 26 in her Vienna apartment, lying face-down on a mattress with splatters of blood on the floor, amid a sea of burning candles, according to the APA news agency.
Investigators believe she died of asphyxiation.
The victim was first reported missing by her employers, who described the au pair as "very reliable", after she had failed to pick up their child from kindergarten.