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 than 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. PLACE YOUR TEXT ADVERT BELOW:>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> How I Made Me =N=2.5 MILLION In 7 Days from 2 Common Product I Imported and how you can do the same. Click HERE FOR FREE Details right now Warning To Men: This 3 Foods Is Killing Your Erection. Click Here To Know Them Secret Fruits that increased my Manhood and makes me last 55 minutes+ all time - Click here to get it Warning: Got Arthritis? 3 Simple Ways To Get Cured And Be Permanently Free From Joint Pains. Click Here To See Them REVEALED!! Reduce Your Blood Sugar Level And Reverse Diabetics Within A Short Period Of Time. Click Here To See How MIDNIGHT,Pressed Mr Kilanko Suddenly Couldn't Urinate. Shrink your Enlarged Prostrate Here! Click Here Now