Mikel Arteta’s side were good value for their triumph, the platform for progress into successive round built on first-half goals from Bukayo Saka and Eddie Nketiah.
Bournemouth substitute sam Surridge force a goal back deep into eight minutes of stoppage to provide the scoreline a glance of reputability and provides Arsenal some late anxious moments, however in reality the Gunners were by far the superior team.
Saka lashed a finish past Bournemouth keeper Mark Travers after only 5 minutes following good work from the outstanding Joe Willock and Gabriel Martinelli, the goalscorer then turning supplier to line up Nketiah for an easy end within the twenty sixth minute.
Bournemouth manager Eddie Howe created six changes from the team that recorded a a lot of-needed Premier League win over Brighton in their last match – however there was still enough expertise in his side to perform with much bigger quality than this mediocre display.
Arsenal, despite that late scare, never seriously threatened to let their lead slip in a controlled performance built round the excellence of Willock and Matteo Guendouzi plus the threat of Nketiah and Saka.
The only blot on their night was a second-half ankle injury that saw defender Shkodran Mustafi taken off on a stretcher.