The Gunners named an unchanged eleven from the 4-2 victory over Leicester City last weekend in the wake of their solid start to the season.
Arsenal went ahead five minutes gone after brilliant piece of individual play from Gabriel Jesus led to a shot by Gabriel Martinelli, which Mark Travers parried only for Martin Odegaard to turn in the rebound.
The Norwegian secured a double five minutes later when he took a cross-field off the foot of Jesus and drilled into the near corner with hints of an offside scrapped by the pitch-side review team.
It was all the Gunners at this point whereas Bournemouth for all intends and purposes provided little to unsettle the visitors, who looked to score more whenever in possession.
Bournemouth started to find some kind of momentum during the latter periods of the half, but the Gunners still enjoyed the lion share of chances as Jesus closed in from a difficult angle.
Manager Scott Parker revived his stall ahead of the second half, as Lewis Cook came on for Ben Pearson and Marcos Senesi sacrificed in place of Jaidon Anthony.
These arrivals brought nothing to trouble Arsenal, who bagged a third goal in the 54th minute after defender William Saliba send a square pass from Granit Xhaka into the top corner.
Bournemouth nearly pulled one back two minutes later from a threatening corner by Marcus Tavernier, but Ben White assisted by Jesus hoofed the threat away from the goal-line.
The Gunners remained compact towards the final 10 minutes and had a fourth through a close-range finish by Jesus, but it was disallowed for offside following a Video Assistant Review.
Arsenal continued to see openings late on and nearly increased their tally from a rasping shot in the area by substitute Eddie Nketiah, but Travers parried and saw the visitors to a comfortable victory.