Despite taking the lead at the Vitality Stadium, we suffered defeat on the south coast to make it five games without a win.
Hector Bellerin seemed to have put the Gunners in control early in the second half, but the hosts hit back with two goals in five minutes.
All the drama came after the interval, the first half was light on clear-cut chances for both sides.
The closest we came in the opening 45 minutes was when Ainsley Maitland-Niles struck the crossbar on 15 minutes. The ball fell to him alone in the area, and after taking a couple of touches to work the shooting opportunity, he bent a right-footed effort which deflected off the bar and away.
Alex Iwobi saw an effort from 20 yards pushed away for a corner as the Gunners, without the attacking flair of either Alexis or Mesut Ozil, looked to take the game to the relegation-threatened hosts.
Bournemouth were well in the game themselves, and were presented with an excellent chance when Maitland-Niles inexplicably passed the ball straight to Jordon Ibe on halfway line. Fortunately for Arsenal though, the winger lost control when bursting into the box.
Danny Welbeck had an effort saved at close range by Asmir Begovic, and Lacazette was unable to control the follow up, but it was Bournemouth who ended the first half in the ascendancy.
Calum Chambers made a superb diving block to deny Dan Gosling inside the area. Bournemouth were hardly playing like a side with just one win from their previous 12 games.