The 23-year-old, who scored 19 goals for Huddersfield in the Championship last season, has penned a six-year deal.
Grant is the Baggies’ ninth signing of the summer with the club taking advantage of the extended transfer window for Premier League and EFL sides.
The former England youth international will return to the top flight after half a season there with Huddersfield in the 2018-19 season.
And Grant believes he is now ready for the step up after flourishing in the Championship.
“I’m 100% confident I can score goals in the Premier League,” said Grant. “I’m a striker and I like to score goals. I’ve scored goals for the last few years and I want to carry on doing that.
“I’ve got the opportunity to play a full season now in the Premier League. I knew it was time to move on in the summer and I always knew I wanted to come here.
“I can’t wait for the challenge. It’s pressure, of course it is. But I like pressure. I want to do well for the team.
“I want to score as many goals as I can as a player and help the team stay in the Premier League. It’s as simple as that.”
West Brom have scored five goals in their opening four games after returning to the Premier League.
And they will hope the addition of Grant can improve their scoring record and ensure they remain in England’s top division.