The Germany U20 international has agreed a four-and-a-half-year deal for an undisclosed fee.
The deal was confirmed by Hammers chief executive Scott Duxbury.
"We are delighted to sign Savio, particularly with the fierce competition for his services across Europe," he told the club's official website.
"He is an exciting prospect, who Gianfranco Zola believes will prove a valuable asset for this club now and in the years to come.
"It is a significant investment and shows our determination to continue moving this club forward."