Serving the final game of his three-match touchline ban, Arsene Wenger watched from the stands as his side lost crucial ground in the race for Champions League qualification.
Bournemouth employed a 5-4-1 formation and their wing-backs got forward well early on, with Adam Smith cutting inside and shooting narrowly wide on four minutes.
His opposite number Ainsley Maitland-Niles came even closer for Arsenal five minutes later when he rattled the crossbar after skilfully creating space for a shot inside the box.
Asmir Begovic saved well from Alex Iwobi and Danny Welbeck in the first half, but without Alexis Sanchez, omitted from the squad as he prepares to leave Arsenal, and the injured Mesut Ozil, who could soon follow him out of the Emirates Stadium, the visitors were not a potent attacking threat.
Iwobi did fashion a clear-cut chance for the Gunners after the break when he exploited the space in Bournemouth's left channel to release Bellerin, who squeezed a shot through the unfortunate Begovic.
Although Arsenal were comfortable in the first 15 minutes after taking the lead, they did not push for a second which may have killed off Bournemouth's resolve.
Petr Cech is one clean sheet away from 200 in the Premier League but surrendered the opportunity to hit that landmark today when he raced off his line and was beaten to Fraser's cross by the outstretched left foot of Wilson, who directed the ball into the vacant net on 70 minutes.
The Cherries completed the turnaround soon after as Wilson held the ball up for Ibe to drill an 18-yard shot through Cech for his first Premier League goal for the club.
In a worrying sign of what could be post Sanchez and Ozil, Arsenal did not have the inspiration to respond themselves and meekly slipped to a fifth game without victory in all competitions, which could leave them eight points adrift of the top four if Liverpool beat Manchester City on Sunday.
Bournemouth's victory lifts them four places up to 13th in the table, four points clear of the relegation zone.
BOURNEMOUTH (5-4-1): Begovic; Daniels (Mousset, 63'), Cook, Ake, Francis, Smith; Cook, Gosling, Fraser, Ibe (Pugh, 86'); Wilson (Afobe, 90')
ARSENAL (3-4-3): Cech; Chambers (Ramsey, 74'), Mustafi, Holding; Bellerin, Wilshere, Xhaka, Maitland-Niles; Iwobi (Walcott, 76'), Welbeck, Lacazette