The 27-year-old makes the switch to The Hawthorns on a four-year deal.
It is the first addition Tony Pulis has made to his squad this summer, having missed out on the signing of Charlie Taylor from Leeds United, following the full-back opting for Burnley over the Baggies last week.
'Buzzing to be here'
West Brom have been linked for move for former Burnley striker Rodriguez during the last 12 months and have now swooped for the England international.
Rodriguez is set to meet his new teammates on Monday, before the team jet of to Austria to begin pre-season training.
On his arrival, he cites that a move to the West Midlands has been in the pipeline for some time and that he is keen to get started for his new club.
"After one or two 'near misses' I'm absolutely buzzing to be here," says Rodriguez, referring to Albion's two previously thwarted attempts to recruit him from the St Mary's Stadium.
"The most important thing for any player is to be wanted," says the Englishman, adding that "Tony Pulis and Albion have made it clear just how much they have wanted me to join. That's fantastic for any footballer to hear."
"I'm genuinely delighted to be joining a club of Albion's stature."
Competition for Rondon
Of the capture, chairman John Williams stated "Jay is a proven Premier League striker, an England international and a player who will give Tony more attacking options."
The arrival of Jay Rodriguez is set to bring fresh competition for striker Salomon Rondon, who in the latter stages of the season was dropped for Hal Robson-Kanu up top, with the Venezuelan struggling for goals.
Rodriguez is also capable of slotting in as a striker, on the wing or in a number ten role, as West Brom have adopted last term.
Having suffered a difficult spell recovering from an ACL last season, Rodriguez will now be keen to capture his best form the 2013/14 campaign when he scored 15 goals.