The Baggies have handed the 30-year-old a two-year deal with an option of a third year and he will wear the number 15 shirt.
He is the marquee signing of Albion's deadline day flurry of activity, and arrives with pedigree in the Championship, having scored 52 goals in his last 84 games in the second tier.
Austin was not in the plans of Saints boss Ralph Hasenhuttl for next season, and was encouraged to find a new club.
Albion first enquired earlier in the window but have now completed a deal for the striker.
It's believed he's taken a wage cut to join the Baggies and play first team football.
He is the second signing Albion have made on deadline day, following Grady Diangana through the door.
But the Baggies are still trying to complete two more deals before the 5pm deadline.
Sporting winger Matheus Pereira is expected to join on loan while a deal for Benfica forward Chris Willock is also being thrashed out.
All of today's signings will be available to make their debuts on Saturday in Slaven Bilic's first competitive home game in charge.