Takumi Minamino impressed in both Champions League group games against Liverpool
The Japan international, who is thought to have a release clause of £7.25m, has signed a four-and-a-half-year deal until summer 2024.
Minamino, 24, has scored nine goals and provided 11 assists in 22 appearances for the Austrian side this season.
"It has been a dream, my dream to become a Liverpool player," he said.
Minamino finalised personal terms and passed a medical on Merseyside on Wednesday.
"To play in the Premier League was one of my targets," he added.
"I think this is the top-class league in the world. I was thinking if my career as a footballer progressed smoothly, someday I would be able to play in the Premier League.
"But I never thought I would be able to play in this team and I'm really happy about it. I'm looking forward to it."
Minamino played in both of his side's Champions League group games against the Reds this season, scoring in the 4-3 defeat at Anfield in October.
Despite that, he would be eligible to play for the European champions in the knockout stages of the tournament following a rule change by Uefa last year.
"From those games I learned this team has such a high level of technique and intensity," he said.
"Their football is of such a high level and I did feel that they were the champions of the Champions League. Feeling that, it makes me feel even happier that I can play in this team. I'm really happy to be part of that team.