Osbourne, 24, has agreed a two-year deal with the Lions, having turned down the chance to extend his contract at Griffin Park at the end of last season.
A product of the Bees' youth academy, Osbourne made 185 appearances for the west London club before announcing in May his decision to quit the club in pursuit of Championship football.
Charlton, Hull and Nottingham Forest had all been credited with an interest in the Southall-born centre-back this summer.
Osborne had previously seen a move to Premier League-bound Reading collapse in the January transfer window after he failed a medical.
But Lions boss Kenny Jackett had no hesitation in handing the defender a deal following a succesful trial and Osbourne will travel to Ireland with his new teammates for the club's pre-season tour, which begins on Monday.