The Cherries had looked set for all three points courtesy of a fine individual goal by Michael Symes six minutes before half-time. But Cox struck to earn a share of the spoils and stun Bradbury’s side.
Fast and direct, second placed-Bournemouth were good value for their lead. They pinned Orient back in their own half for the opening 20 minutes, with goalkeeper Jamie Jones making fine saves from Anton Robinson and Liam Feeney.
Symes gave them the lead with a fine goal. Picking up the ball deep inside the Orient half, then running unchallenged before planting a measured shot into the right-hand corner.
Orient, beaten just once in 17 league and cup outings, looked dangerous on the break and Jonathan Tehoue forced a fine save from Shwan Jalal in the second half.
Danny Ings twice missed chances to seal the points for the Cherries, but Orient manager Russell Slade’s decision to throw on all three substitutes paid off late on.
Cox looped a cross-shot over the head of Jalal to steal a point for the O’s with barely 15 seconds of normal time remaining.