The 24-year-old had been at Bristol for four seasons and before that was part of the now defunct Richmond professional side.
Best had 12 Premiership starts for the Shoguns last season and is well-acquainted with his new club, having appeared against them on five occasions during his Zurich career.
"Lee is a player who has considerable Zurich Premiership experience, and has also represented England at A-level," said Bath head coach John Connolly.
"He joins us initially as World Cup cover, but there is also a view to extend that stay. I am sure that he will make a positive impact when the club is without the established international players."