Nicolas Jackson and Raheem Sterling both hit the post but another disjointed performance saw the Blues fail to score for the second game on the trot.
After losing 1-0 to Nottingham Forest in their last league game, Chelsea dominated possession in the first half with Nicolas Jackson striking the foot of the post in the 14th minute and Conor Gallagher wasting a glorious opportunity just after the half-hour mark.
The visitors had the ball in the net early in the second half as Raheem Sterling struck the angle of post and crossbar with a free kick and Levi Colwill blasted home the rebound, but the goal was ruled out for offside.
Bournemouth had some decent chances and keeper Neto pulled off a superb one-handed save to deny Chelsea substitute Cole Palmer as they hung on for a draw that moves them up to 15th spot on three points, two points and one place behind Chelsea.
Additional reporting by Reuters.