The 23-year-old has been strongly linked with Albion in recent days - with the Baggies and Birmingham City reported to have matched his release clause of £300,000
However it is the Baggies who have won the race for the striker's signature.
The forward, who has netted five goals in seven games for the Ammies so far this season is now in line to make his debut against Burnley on Friday.
And he has revealed the move happened quickly and he is eager to show the Albion fans what he is all about.
He said: “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.
“I believe I have an eye for goal and I’m confident I can score here and push the team forward. I can’t wait to get started, meet my new team-mates, settle into the area, and show the fans what I’m all about.”
The striker, who can also play as a winger, joined Salford back in 2019 and has so far netted 30 times in 97 starts for the club.
Baggies boss Steve Bruce added: "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.”