The winger has moved to The Hawthorns after the two clubs agreed an £800,000 fee with add-ons.
Grosicki has agreed an 18-month deal and is registered in time to be in squad to face Luton tomorrow.
The 31-year-old's capture could be the start of a number of number of deadline day signings for the Baggies.
Albion are also believed to have made an enquiry to Watford in an attempt to take Portuguese midfielder Domingos Quina on loan.
And another striker remains high on the list of priorities.
Slaven Bilic is due to hold a press conference at midday today.