Arsenal extend lead at top of Premier League
Goals from Gabriel Magalhães and two from the returning Declan Rice sent Arsenal six points clear of second-placed Aston Villa and seven ahead of Manchester City in third spot.
Evanilson gave an impressive Bournemouth the lead in the 10th minute on England's south coast, stroking home after an uncharacteristically sloppy pass from Gabriel.
However, the Brazilian defender made up for his error just five minutes later, converting after Noni Madueke's drive to the byline following a set-piece routine, making it 1-1.
Declan Rice, who missed Arsenal's thumping 4-1 win against Villa in midweek due to a knee problem, side-footed the Gunners ahead in the 54th minute.
Rice then gave the visitors some breathing space 20 minutes from time, producing a low finish from substitute Bukayo Saka's cutback for 3-1.
Junior Kroupi brought Bournemouth back into the game with a stunning right-footed finish that flew into the top corner, but Arsenal held on for a fifth straight Premier League win, keeping them firmly on course for a first English title since 2004.