The Gunners gave a run to Eddie Nketiah, who recently returned from a loan spell at Leeds United, while the other key changes saw Matteo Guendouzi and Joe Willock recalled to the midfield.
Arsenal wasted little time to hit the front when Bukayo Saka fed off a through ball from Gabriel Martinelli to blast beyond Mark Travers from the near post.
The Gunners should've had a second goal on 21 minutes after Willock got on the end of a close-range delivery before firing wide of Travers's goal.
Nketiah announced his return with a simple finish from inside the area following a low ball into the six-yard box by Saka, with the Video Assistant Referee approving on 26 minutes.
Bournemouth put together a dangerously looking move on 38 minutes for Harry Wilson, who forced Gunners shot-stopper Emiliano Martinez to a save after meeting Granit Xhaka's cross.
The start to the second half was evenly balanced until the 59th minute when Martinez raced off his goal-line to deny Dominic Solanke whilst crashing into his own teammate Shkodran Mustafi, who was stretched off following a nasty collision with his goalkeeper.
The Cherries could've pulled a goal back on 65 minutes from a corner on the right side, but Steve Cook headed wide of the near post after rising high in the area.
Eddie Howe's side found their goal in added time when Sam Surridge flicked home a diagonal ball from Simon Francis, but it was too little too late as the Gunners held on for the victory.