The Englishman penned a four-year contract with his new club.
According to Sky Sports, the Baggies paid an initial £12m for the 27-year old, but the fee could rise to £15m.
The Englishman scored 35 goals in 125 appearances with the south-coast club.
The Baggies chose an unorthodox way of telling their fans that Rodriguez had signed for the club.
They responded to a question on Twitter asking when their first signing would be, Albion then replied with a video clip of Rodriguez saying “yes we have".
Navy blue and white stripes. They suit you, @JayRodriguez9!#WelcomeJay#WBApic.twitter.com/sYc3AutSwz
— West Bromwich Albion (@WBA) July 2, 2017