Hector Bellerin gave Arsenal the lead in the second half, before a four minute double from the Cherries gave them the win.
Callum Wilson grabbed the Bournemouth equaliser before Jordan Ibe scored a timely first goal for the club as the Gunners' struggles on the road continued.
No Sanchez an early talking point
The morning of the game sparked more rumours of Alexis Sanchez's potential destination in January and with a move to Manchester United supposedly edging ever closer he was left out of the Arsenal squad entirely for their trip to Bournemouth.
It was a selection that displayed the alarming lack of quality available to Arsene Wenger - who was also missing Mesut Ozil through injury - meaning the Gunners' boss was forced to field an XI which from the outset looked as though it would struggle for goals - and indeed that was the case.
Alexandre Lacazette, a player who has been low on confidence of late started up top - and with Sanchez departing the Frenchman now has an even bigger onus on him to become Arsenal's main goal scorer, something he couldn't manage on the day.
Arsenal come close in uninspiring first half
The Gunners lack of confidence in attacking areas was evident throughout the game with Danny Welbeck and Alex Iwobi looking timid whenever they got in and around the penalty area.
The first half was summed up from an attacking point of view for Arsenal by Ainsley Maitland-Niles going closest. After eight minutes he received the ball in the box - twisting and turning - then striking an effort against the cross bar.
Welbeck also came close to opening the scoring, as he was slipped in by Lacazette but Asmir Begovic came out well to deny the winger.
Aside from that the visitors were restricted by a compact Bournemouth defence but Shkodran Mustafi did have two opportunities from first half corners, heading wide on both occasions.
An Arsenal defence including Mustafi, Rob Holding and Calum Chambers did well against Chelsea in midweek but they always had the potential to look shaky and they did at times against Bournemouth. Ryan Fraser found space in the box with five first half minutes remaining, only to be denied well by a Chambers block as the Gunners allowed an easy ball to come into the area from Ibe.
Bournemouth were perhaps quite unfortunate not to be awarded a penalty before the break as Iwobi looked to have moved his arm towards the ball from a corner. Referee Kevin Friend had a brilliant view but turned down strong appeals from those in red and black.
Bellerin draws first blood
It didn't take long for Arsenal to finally find the net in the second half and the breakthrough came from Bellerin on 52 minutes.
Iwobi played a lovely ball in behind to set the Spaniard free, before he raced into the penalty box and squirmed the ball through Begovic to find the net and grab his second goal in successive Premier League games.
Cherries hit back with quick double
With 20 minutes remaining Bournemouth found a deserved equaliser. The Gunners had warnings prior to the goal, with Ibe and Fraser forcing stops from Petr Cech, but the Arsenal 'keeper then did poor as Eddie Howe's side made it 1-1.
Fraser whipped a lovely ball into the area, and Wilson stole in ahead of two Arsenal defenders and Cech who came racing out of his goal as the Bournemouth striker poked the ball home.
Four minutes later the hosts found themselves in front. A ball over the top found substitute Lys Mousset who was then able to tee up Ibe as he fired under Cech and made it 2-1.
Pressure continues to grow for Arsenal
The Gunners defeat means they are still yet to win in 2018, and have only take maximum points from three out of 12 Premier League fixtures this season.
For Arsenal, their season is falling away without a trace. Knocked out in the FA Cup third round, out of the top four in the league, and their best player in Sanchez leaving - Wenger has a massive job on his hand if he is to somehow rescue something from this season.
Before Manchester City's game with Liverpool, the North London side are now 23 points off the league leaders and five points off a Champions League place; troublesome times for Arsenal.