Coming in the 47th match of the campaign, it equals the Premier League record for the longest start to a season without a 0-0.
How the match unfolded
Chelsea started brightly, with Conor Gallagher, captaining the side after Ben Chilwell was only named among the substitutes, unable to hit the target from close range before Nicolas Jackson struck a low drive against a post.
Bournemouth, playing largely on the counter-attack, came close to opening the scoring after Sanchez denied Dango Ouattara from point-blank range.
The best chance of the first half fell to Gallagher, but his low shot from 10 yards out was kept out by Neto.
Chelsea remained in control in the second half and hit the woodwork for a second time on 50 minutes when Raheem Sterling’s free-kick crashed against a post.
The ball bounced down on to the goalline before falling to Levi Colwill, who tucked in to score, only for his strike to be disallowed for offside.
Goal Decision System of Sterling free-kick
Bournemouth survived a goalmouth scramble after Neto clawed away Colwill’s close-range effort, and went close at the other end when Philip’s free-kick was deflected wide by Gallagher.
It was the hosts' best spell of the match, as Dominic Solanke then forced another save from Sanchez, this time with his legs.
Chelsea came back again, with Jackson and substitute Cole Palmer having shots saved by Neto, but neither side could find a winner.
The Blues have registered 35 shots over their last two matches, but have been unable to score. Indeed, their five goals so far this season are their fewest after five matches of a season since 1995/96, when they finished 11th.
With only one win so far, Mauricio Pochettino's side drop to 14th, on five points, two more than Bournemouth, who are still awaiting their first win of the season, in 15th.