Sako had the opportunity to extend his stay at Crystal Palace in the summer, but the Mali international rejected the offer of a new contract from the Eagles.
The 30-year-old has remained without a club for a number of months, but the Baggies have now confirmed that they have pushed through a deal for a player who was previously regarded as a cult hero at fierce rivals Wolverhampton Wanderers.
Head coach Darren Moore told the club's official website: "He brings great experience and adds greatly to our goal threat across a range of positions. We're delighted to get him.
"He brings competition to the group and a know-how of this division. He's been a credit to himself while he's been training with us and I can vouch for the fact that the way he has committed to us has been exceptional."
Sako moves to The Hawthorns with West Brom currently sitting at the top of the Championship standings.