Mikel Arteta's side still had ambitions of competing on multiple fronts this season, but they were second best for long spells against an in-form Bournemouth team that arrived unbeaten in 11 league matches.
The visitors made a confident start and took the lead just after the 15-minute mark. Eli Junior Kroupi finished from close range at the far post after a low cross from Adrien Truffert was deflected kindly into his path.
Arsenal struggled to find rhythm in the first half but were handed a route back into the game midway through the opening period. Gabriel Magalhães saw his effort strike the hand of Ryan Christie, and the referee pointed to the spot.
Viktor Gyokeres stepped up and converted the penalty with composure to level the score.
Despite that equaliser, Bournemouth continued to look the more threatening side before the break. David Raya was forced into a save from Evanilson, while Alex Scott fired over as the visitors carried the greater attacking intent.
The pattern continued after the interval, with Bournemouth controlling possession and unsettling Arsenal. The home side had the ball in the net through Gyökeres, but the goal was ruled out for offside. Declan Rice tried to inspire a response, forcing Dorde Petrovic into a strong save from distance.
However, Bournemouth regained the lead in the 74th minute when Alex Scott finished confidently after Evanilson nudged the ball into his path. Arsenal pushed forward late on, with Gabriel Jesus going close, but they lacked the cutting edge to break through.
The result lifts Bournemouth into the top half and underlines the impressive work of Andoni Iraola, while Arsenal are left needing a strong response as the title race intensifies.