Antonio rose highest at the back post to head in Gokhan Tore's cross and decide a game of few chances.
Hammers striker Enner Valencia drew a good save from Artur Boruc in the first half, while Adrian brilliantly tipped over Joshua King's drive.
Harry Arter was sent off for two yellow cards as Bournemouth ended with 10 men.
West hamhave sold over 50,000 season tickets and the club boast the highest number of Season Ticket Holders in London, and the highest percentage of Season Ticket Holders in the whole of the Premier League.
The Club’s increase in capacity from 35,000 at the Boleyn Ground to 60,000 at the former Olympic Stadium sees the hammersd almost double their home crowd.