The 23-year-old put pen to paper on a three-year deal after an impressive start to the season where he has netted four goals in six in League Two appearances for Salford.
Baggies boss Steve Bruce told West Brom's website: "Brandon is a young, hungry, forward who has shown great desire and a strong mentality to earn his chance at this club.
"He is a strong, pacy player, who has a good record in front of goal, and I am sure he is excited to show people what he can do at this level.
"Brandon further strengthens our attacking options and his versatility to play across our front line will, I'm sure, be of great value as we continue to take on a busy Championship schedule."
Thomas-Asante added: "I'm over the moon to be joining such a big club.
"It all came about quite quickly but as soon as I spoke to Steve Bruce I realised this was the right place for me."
Thomas-Asante becomes Bruce's fifth summer signing as he looks to bring more attacking threat into his side prior to the end of the transfer deadline.