The 22-year-old, who is is reported to have cost the Saints €12.5 million, becomes Ronald Koeman's ninth signing of the transfer window and his second of deadline day.
Shortly beforehand, the south-coast club had announced the arrival of defender Toby Alderweireld on loan from Atletico Madrid.
Mane, a Senegal international, spent two years with the Austrian side after leaving French club Metz and has now put pen to paper on a four-year contract at St Mary's Stadium.
"I saw him playing last season and I was impressed with his qualities and with the physicality that he showed.
"He can play in different positions in the attack – he can play on the left or right and also as a No.9.
"It is incredible how many goals he has scored from his position and I hope that he will do the same for us."