An accident waiting to happen' - Arteta rues Saliba's red card
The France centre-back was sent off following a VAR review in the 30th minute for bringing down Evanilson and denying him a goalscoring opportunity.
Despite digging in, Arsenal were unable to maintain their unbeaten start, as Ryan Christie and Justin Kluivert's second-half goals subjected them to a first defeat of the season.
It is the first campaign since 2011-12 that the Gunners have had as many as three players sent off in their opening eight matches of a Premier League season.
In fact, since Arteta's first game in charge of Arsenal on Boxing Day 2019, they have been shown 18 red cards in the top flight, at least five more than any other side.
While they managed to hold onto results against Brighton and Manchester City after going down to 10, Arteta admitted it was going to cost them eventually.
"It's very difficult to win in the Premier League with 10 men for 60-70 minutes, it's just an accident waiting to happen. We have had to go through it in three games and that cost us the game," he told BBC Match of the Day.
"We had the big chance and the moment in the game to score and have a different result, and then we conceded a goal. We try to stick to the rules. The three [red cards] have been different ones and three very different outcomes.
"The team tried, the effort with 10 men was fantastic, but today it was not enough to win the game. It took us a while to understand what we had to do in the first 15 minutes to start with, then we grabbed the game, but after 30 minutes, the game changed."
Bournemouth secured just their second-ever Premier League victory over Arsenal in 15 meetings, and their first since January 2018, when they won 2-1.
Andoni Iraola's substitutes did the damage for the Cherries, with Kluivert both scoring and assisting in a Premier League game for the first time.
Iraola was delighted by the maturity shown in his team's performance as they bounced back from a disappointing loss to Leicester City before the break.
"It was quite a mature win because it is a tricky situation [playing against 10 men]," Iraola told Sky Sports.
"We started well for the first 30 minutes and then the red card changes the game. We had to control the game, attack but not lose our minds, as even with 10 men, set plays, free-kicks they could be a threat. Luckily for us, we took the benefit of having one man more.
"Before half-time, we missed a couple of chances and at the beginning of the second half we managed not to finish, so we had to have patience to take the benefit of having one more player without losing our heads.
"We were playing very well the first 30 minutes. It was a very level game, which is very good when you face this opposition. It became easier with 11 v 10 but everything is not done, you have to finish the job, and we did it."