The Irishman, 23, will join the Exiles at the end of the season on a two-year deal.
"Eoin is an exciting, young Irish talent," director of rugby Brian Smith told the club's official website. "He is a big, physical centre with great feet and distribution skills, so he will fit in well with the style of rugby we play.
"The form of our centres this season has been superb so competition for places next season will be fierce which is what every coach is after."
Griffin signed his first professional contract with Connacht in 2009.