The 22-year-old has signed a four-year deal with the Black Country outfit, after demanding a move away from Hillsbrough.
"Chris is a player whose ambitions match our own. He wants to play at the top level and we're delighted he's chosen our club as a vehicle to take him there," said Baggies boss Tony Mowbray.
"He has undoubted, proven quality at this level and brings a cultured left foot to the squad. He will supply plenty of ammunition for our strikers and chip in with a few goals himself."
Young Manchester City striker Miller has joined on a season-long loan with a view to a permanent deal.
"Ishmael will give us a different option up front. He's a big, quick, powerful lad who is looking for an opportunity to play more regularly," added Mowbray.
"He will increase competition for places with the other strikers we've already got at the club."