It was a manic first half as the Cherries took the early lead through David Brooks as he finished after a long throw, before Cole Palmer equalised from the spot nine minutes later with 15 minutes on the clock.
Chelsea took the lead through Enzo Fernandez, but Justin Kluivert struck to equalise four minutes later from another long throw.
While the second half had much of the energy of the first, there were fewer clear-cut chances for the two sides and neither could find the deciding goal and the spoils were shared - although the Cherries will feel they could have won it with an excellent Enes Unal chance late on.
That takes the Cherries to ten games without a win but they impressed after the weekend’s disappointment at Brentford, taking Bournemouth to 23 points at the halfway stage of the season.
Andoni Iraola made four changes to his team that was well-beaten at Brentford.
Bafode Diakite, Adam Smith, Lewis Cook and Junior Kroupi all dropped out, with James Hill, Brooks, Kluivert and Evanilson coming back into the team.
Diakite, Smith and Kroupi dropped to the bench, while Cook was ruled out with a neck knock, as Iraola sought a response from his side which fell to defeat against the Bees.
It was an energetic start from the Cherries, and they were rewarded with the opening goal after six minutes.
Brooks first played Evanilson in and his shot was blocked out for a throw in, which was taken by Antoine Semenyo and flicked on at the near post.
The Welshman then met it with his head from the centre of the six-yard box, it was saved by Robert Sanchez but Brooks was able to scramble the ball into the back of the net.
But they weren’t in front for long, as Chelsea were awarded a penalty after a VAR check for a Semenyo challenge on Estevao.
Palmer stepped up to cooly convert into the bottom corner past Petrovic and the game was level again.
The Cherries kept coming, though, and an Evanilson header was caught by Sanchez before Brooks powered a shot straight at the keeper when played in by Evanilson.
But Chelsea hit the front after 23 minutes when Alejandro Garnacho passed to Fernandez at the top of the box, he turned well to take Alex Scott out of the game and then powered into the top corner.
Bournemouth went close through Marcus Tavernier as he volleyed just over the bar when Brooks’ shot was blocked out to him.
Iraola’s side were back level from another long throw as Trevor Chalobah flicked it on at the front post and Kluivert was there to touch the ball home.
Malo Gusto shot well over the bar from distance, before Alex Jimenez fired over from a tight angle at the other end.
The chances continued to come for both sides as Estevao headed a Garnacho cross pass the post and Truffert volleyed into a crowd of bodies as a long throw was headed out to him.
Semenyo flicked a Brooks cross towards goal that Sanchez saved but did not fully deal with as Evanilson threatened, but the Blues were able to scramble clear.
Estevao had a shot blocked by Truffert and scooped up by Petrovic in the final chance of a breathless first half.
The Cherries started the second half brightly, and threatened again from a long throw, although Chelsea were able to prevent a Bournemouth player from getting the telling touch.
Semenyo shot over the bar before the Cherries had a penalty appeal turned away after a short VAR check for a handball from Gusto.
Chelsea had a good opportunity on the counterattack from the resulting corner, as Pedro Neto crossed for Liam Delap, but he headed over when well placed.
Petrovic saved well from an Estevao shot from range after Evanilson had lost the ball in the Cherries half, before Fernandez shot over the bar from the top of the box.
Amine Adli replaced Brooks with 12 minutes to play, before Enes Unal was introduced for Evanilson with eight minutes left.
Kluivert was replaced by Smith shortly after Unal volleyed over from about 12 yards out when he was found by an Adli cross.
Chelsea tried to apply the pressure late on but the Cherries held firm to win a point on the road at Stamford Bridge.
**Chelsea:** 1, R. Sanchez; 27, M. Gusto; 29, W. Fofana; 23, T. Chalobah; 34, J. Acheampong (James 46’); 8, E. Fernandez (c); 25, M. Caicedo (booked, 4’); 49, A. Garnacho (Neto 46’); 10, C. Palmer (Joao Pedro 63’); 41, Estevao (Gittens 90+3’); 9, L. Delap (Andrey Santos 87’).
**AFC Bournemouth:** 1, D. Petrovic; 3, A. Truffert; 5, M. Senesi; 23, J. Hill; 20, A. Jimenez; 8, A. Scott; 16, M. Tavernier (booked, 84’); 24, A. Semenyo; 19, J. Kluivert (Smith 90+3’); 7, D. Brooks (c)(Adli, 78’); 9, Evanilson (Unal 82’).