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West Brom did not disclose how much they have paid to sign him, but British media reports said the fee was in the region of ÂŁ15 million ($19.2 million, 16.3 million euros).
A graduate of Nottingham Forest's academy, Burke left Forest for Leipzig last August and helped the club achieve a surprise second-place finish in the Bundesliga, but featured only sparingly.
"I feel like it is a time for me to move on and play more games," Burke told the West Brom website.
"I'm here, I want to impress and I want my name to be first on the team sheet."
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Burke is West Brom's fifth acquisition of the transfer window after Gareth Barry, Jay Rodriguez, Zhang Yuning and Ahmed Hegazi.
He could make his debut in Sunday's home game with Stoke City.
"We're really pleased to get him in -- the lad has got a tremendous talent," said West Brom manager Tony Pulis, whose side have won their opening two league games.
"He's got a fantastic, raw ability, but he has now also got the experience of going away from the comfort of his own area to a foreign country and experiencing life.
"He's gained the experience of playing for a top club in the Bundesliga with top coaches. That will have broadened his horizons and been good for his development."