Ramsay moves to the Hawthorns from his job as head coach of Minnesota United. With the MLS franchise, Ramsay led them to consecutive semifinal places.
The former Manchester United set-piece coach succeeds Ryan Mason at West Brom.
"Throughout our recruitment process, Eric consistently exhibited the qualities to lead and succeed in the world's most demanding football environment," Albion chairman Shilan Patel told the club's website.
"In his time as a head coach, his teams have been characterised by intensity, organisation, and adaptability – all critical for the sustained success and growth that we envision in the short and long term."
Ramsay, 34, has signed a deal to 2028.
"Eric's leadership qualities and attention to detail are key traits that stuck out to us, and he has demonstrated an ability to adapt his playing style to get the best from players at his disposal without compromising his football philosophy," said Albion's president and sporting director Andrew Nestor.
"His recent work as head coach at Minnesota over the past two seasons has reinforced this, having led a squad to perform well beyond expectations against clubs with greater budgets."