The forward completed the full game while his counterpart was withdrawn after the hour mark.
Key Moments
Two assists from Semenyo's long throw-ins led to strikes by Brooks and Kluivert.
Cole Palmer and Enzo Fernandez found the net for Chelsea in a frantic first half.
The second half's clearest chance fell to Fernandez, who failed to convert.
Bournemouth forward Antoine Semenyo, reportedly close to a big-money transfer, was at the heart of the action, providing two assists and conceding a penalty in a dramatic 2-2 stalemate against Chelsea.
Despite being on the verge of a lucrative switch, the 25-year-old attacker started for his club. He utilized his long throw expertise to great effect, creating scoring opportunities that were finished by Brooks and Kluivert.
The deadlock was broken early when Semenyo's first long throw caused havoc, allowing Brooks to score from close range in the sixth minute.
The home side equalized and went ahead following a foul by Semenyo on Estevao in the box. The match official awarded a penalty after a review by the Video Assistant Referee system.
Palmer calmly dispatched the penalty, placing it perfectly into the corner even though the keeper dove the right way. Fernandez made it 2-1 with a stunning strike into the top corner midway through the first half.
Kluivert leveled the scores again shortly after, tapping home from another precise long throw delivered by Semenyo.
The visitors finished the first period on top, forcing Sanchez into several saves from their 14 first-half efforts.
At halftime, Chelsea's manager made two changes, introducing captain Reece James and Pedro Neto for Josh Acheampong and Alejandro Garnacho.
A potential handball incident involving Gusto was quickly reviewed and not given by the video officials. Former Chelsea goalkeeper Djordje Petrovic then produced an excellent save to deny Estevao from distance.
The best chance of the second half fell to Enzo Fernandez in the 77th minute, but he blazed his shot over the bar from inside the box.
Moments earlier, Palmer was substituted for Joao Pedro, a change that was met with disapproval from some home supporters. Although they enjoyed more possession late on, the hosts were unable to craft a winning goal.
In injury time, Bournemouth substitute Enes Unal volleyed over from 12 yards out after a cross from Amine Adli.
The result means Chelsea have managed just one victory in their last seven league outings. Bournemouth's winless streak now stretches to ten matches across all competitions.
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