The forward leaves the Terriers in a deal believed to be worth at least £15m just 21 months after joining from Charlton Athletic for £2m.
Grant was Town's top scorer last season, racking up 19 goals to help push them to safety in their battle against relegation from the Championship - and his four goals in 13 Premier League games after joining the club in January 2019 made him their top scorer that season, too.
West Brom have been in the market for a forward since confirming their automatic promotion to the Premier League in July by finishing second in the Championship.
