David Vaughan had put Blackpool in front but Albion replied through Odemwingie and James Morrison only for Gary Taylor-Fletcher to level for the Seasiders.
But then Odemwingie struck with three minutes remaining to move Albion away from the relegation zone.
Blackpool, fresh from their midweek triumph over Liverpool, played with their customary freedom with the in-demand Charlie Adam in fine form in midfield.
But they were left to rue two missed chances by the normally deadly DJ Campbell.
Source: PA
Source: PA