The Senegal international was a wanted man across Europe, but Ronald Koeman won the race to sign the 22-year-old just as the deadline approached.
Mane scored 45 goals in 87 appearances for the Austrian champions after joining them from French club Metz two years ago.
He has also scored three goals in 15 appearances for his country, for whom he played in the Olympic Games in London in 2012.
Koeman said: “Right from the first meeting I had with Les Reed, when it was clear that we would have to change a lot, Sadio was on our list of potential players.
“I know the player because last season Ajax Amsterdam played against Red Bull Salzburg in Europe, and I saw him playing 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 side and also as a number nine. It is incredible how many goals he has scored from his position, and I hope that he will do the same for us.
“With the way that we play and the fact that I know him very well, this is a good step forward for him.”