The Hammers almost opened the scoring within a minute, but Axel Disasi’s header from a Jarrod Bowen corner was saved, and the centre-back’s follow-up ended up being deflected over.
Crysencio Summerville then sent his attempt from a perfectly-weighted Taty Castellanos pass straight at Đorđe Petrović, before the visitors settled into proceedings and started to have the better of possession, although Rayan’s speculative shot was held by Mads Hermansen.
In notable highlights of an otherwise dull opening act in the capital, Soungoutou Magassa’s attempt was blocked before Marcos Senesi’s header was routinely saved.
And Castellanos’ header from a Bowen corner was deflected behind by James Hill, before the Hammers captain sent an effort of his own harmlessly wide.
There was then an ambitious strike at either end shortly before the break, although Alex Scott’s attempt soared straight towards Hermansen and Summerville’s sailed off-target.
Magassa saw another hopeful shot blocked after the restart, and the midfielder was then unable to convincingly control an Aaron Wan-Bissaka ball into the box.
At the other end, Rayan caused a real scare soon after, gliding forward and curling an effort wide off the post.
David Brooks then had a shot deflected behind and another blocked shortly after being introduced from the bench, before an acrobatic Castellanos shot sailed off-target and El Hadji Malick Diouf slid in to deny Rayan.
The teams stepped up their search for a winner as the endgame approached, with an instinctive Callum Wilson effort and Scott's drive being saved at either end, before the latter deflected Summerville’s strike over.
And despite being a constant thorn in Bournemouth's side, Summerville was denied yet again as the game moved into six minutes of added time, this time by Álex Jiménez.
That said, the best chance of a West Ham winner came from Bowen with 90 seconds to play, with the World Cup hopeful firing over under pressure from a sliding Adrien Truffert challenge.
With neither team able to find that killer touch despite Marcus Tavernier looking bright off the bench, West Ham extended their record to one defeat from their last eight matches across all competitions.
Alarmingly, though, the last three of those have ended level after 90 minutes, and the Hammers could still end tomorrow’s play up to five points adrift of 17th-placed Nottingham Forest.
Meanwhile, Bournemouth are unbeaten in seven and remain hopeful of a European charge.
Flashscore Man of the Match: Alex Scott (AFC Bournemouth)
