Antonio rose highest at the back post to head in Gokhan Tore's cross and decide a game of few chances.
The goal came eight minutes after Bournemouth's Harry Arter was sent off for a second bookable offence.
Cherries goalkeeper Artur Boruc saved well from Enner Valencia in the first half, while Adrian denied Joshua King.
A club-record 56,977 supporters were in attendance as West Ham played their first league match at their new home, having left Upton Park at the end of last season.