With title rivals Manchester City and Liverpool not in action until Sunday afternoon, Arsenal had the chance to leapfrog both of their adversaries into first place in the top-flight table by overcoming the Cherries in the 5.30pm kickoff.
However, Arsenal once again fell victim to their own indiscipline, as Saliba received the first red card of his career inside the first 30 minutes for a last-man foul on Evanilson.
The Frenchman's dismissal marked the third sending-off Arsenal have received in eight games this Premier League term, following Declan Rice's red against Brighton & Hove Albion and the ejection of Leandro Trossard - whose terrible backpass triggered Saliba's panicked challenge - against Manchester City.
Arsenal held out for a short while with 10 men, but Bournemouth eventually battered down their defences when Ryan Christie finished off a neat corner routine, before Justin Kluivert struck from the spot after David Raya brought down Evanilson.
Arteta suggests Arsenal will not appeal Saliba red card
Saliba's foul on Evanilson was initially deemed worthy of a yellow card by referee Rob Jones before he changed his mind upon viewing the pitchside monitor, and Arteta cut a down-the-line tone when asked about the incident after the game.
"Well, it was a decision made on the pitch, that decision was changed and now I don't think the decision is going to change twice. So, decision made," Arteta said in his post-match press conference.
"It's a split second decision you have to make, the striker is on his back. Obviously, he doesn't expect the ball and then suddenly it's coming his way. He's made a decision and that's it, the referee has decided to give a yellow and then a red card."
When asked whether Arsenal would appeal Saliba's sending off, Arteta suggested that the Gunners would not bother, adding: "The decision has been made, so that's it."
Saliba's sending-off has triggered a one-match suspension for the Frenchman, who will serve his domestic ban in next weekend's crunch clash with current leaders Liverpool at the Emirates, but he is available for Tuesday's Champions League tie with Shakhtar Donetsk.
Who will replace Saliba for Arsenal against Liverpool?
When Arsenal lose Saliba, they lose one of the most composed, collected and - usually - level-headed centre-backs on the planet, one who exudes confidence on the ball and can time his tackles like no other.
The Frenchman's ban comes at a time where Arsenal are already sweating over the fitness of two possible replacements, Jurrien Timber and Takehiro Tomiyasu, the latter of whom may be out for a while due to a new knee setback.
However, Timber may be able to return for the Liverpool game, where either he or Ben White - who played right-back on Saturday after recovering from his own knee issue - could join forces with Gabriel Magalhaes in the middle.
White could also shift centrally even if Timber is not fit, allowing Thomas Partey to deputise as an emergency right-back, and Arteta's only other solution would be to use another left-footed centre-back in defence.
Riccardo Calafiori and Jakub Kiwior would be the main contenders - either Oleksandr Zinchenko or Myles Lewis-Skelly could be drafted in at left-back - but Kiwior was largely at fault for Bournemouth's second on Saturday and was hooked after coming on as a substitute.