The Cherries are yet to win in the league under Andoni Iraola, while Mauricio Pochettino’s Chelsea have one win to show from their five league games.
Chelsea had the better chances with Nicolas Jackson and Raheem Sterling hitting the woodwork and Levi Colwill having a goal disallowed for offside.
New Blues signing Cole Palmer had a chance to win the game late on but shot straight at Neto.
Bournemouth had opportunities to win too with Dominic Solanke forcing a Robert Sanchez save and heading wide.
The final ball was lacking and strikers misfired in a game where both sides were short of confidence and defences were on top.
Chelsea, for all their expensive, expansive recruitment, had six substitutes with no Premier League experience and academy product Conor Gallagher wearing the captain’s armband.
The Blues’ £100m midfielder Moises Caicedo missed out with a knee injury from his international duty with Ecuador.
Their first major chance came when Mykhailo Mudryk, making his first Premier League start under Pochettino, found Jackson, who struck the base of the post from the edge of the area.
Raheem Sterling was not quite as electric as in previous weeks with none of his four shots testing the goalkeeper.
Captain Gallagher should have done better when he wriggled free from 10 yards out but his shot was too central and Neto got down low to save.
Probably their best moment of the game saw Sterling’s free-kick hit the angle of post and bar and bounce on the line without fully crossing it. The ball reached Colwill, who fired home – but the Blues defender was offside when the free-kick came in so had his effort ruled out.
Jackson almost won it late on but did not get enough direction on to his header from substitute Ben Chilwell’s cross and Neto saved.
If the Blues can improve their finishing, results will pick up – but for now they sit 14th.
“I cannot lie, I think we are missing something in the last third – more calm, more relaxed, to be more clinical in front of goal,” said Blues boss Pochettino.
“That is about giving confidence and trust in your offensive players. I think we were creating chances, fighting, I think it’s about the process.”