The Gunners came into the game on the back of two costly defeats by Stoke and Liverpool, but made the perfect start when Welbeck headed them in front after six minutes.
Record signing Alexandre Lacazette doubled the lead on 27 minutes, bending a shot past Asmir Begovic from the edge of the area.
Welbeck then added a third just after half-time, scoring from a tight angle into Begovic’s far corner.
Alexis Sanchez, who was close to joining Manchester City on transfer deadline day, started the day on the bench, but came on for Welbeck with 15 minutes remaining.
The Cherries failed to register an attempt in the first half and Jermain Defoe’s header against the post after half-time was the best it got for the visitors.
Arsenal were reduced to 10 men for the final 10 minutes, when substitute Francis Coquelin was forced off with a hamstring injury but Bournemouth failed to take advantage.
Eddie Howe’s side are only off the bottom of the Premier League on goal difference and have lost their opening four games in a season for only the third time.
Arsene Wenger’s side were woeful in a 4-0 defeat at Anfield two weeks ago, which followed a 1-0 loss at Stoke.