The Gunners had lost to both Stoke and Liverpool heading into the international break but they were never in trouble here as Welbeck inspired them to a 3-0 win against a poor Cherries side.
Welbeck opened the scoring early on with a mistimed header before setting up Alexandre Lacazette and then completing his brace with a well-taken second.
Manager Arsene Wenger recalled both of his summer signings in Lacazette and Sead Kolasinac and both impressed as Alexis Sanchez was only named on the bench.
After losing 4-0 at Liverpool last time out, Welbeck settled any nerves with an early goal -- getting enough on Kolasinac's cross to divert the ball past Asmir Begovic with six minutes gone.
Kolasinac, operating as a wing-back rather than in Wenger's back three, was causing problems for the Cherries defence and another marauding run culminated in Granit Xhaka stinging the palms of Begovic with a low drive.
Mesut Ozil was next to test the former Chelsea goalkeeper, bending a free-kick towards the corner only for Begovic to push it clear.
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The second goal came just before the half-hour mark, Ozil playing a pinpoint pass to Lacazette, who exchanged passes with Welbeck before finishing well.
Arsenal's club-record signing could have had a second soon after but Begovic recovered after hitting a clearance straight at Welbeck to dive at the feet of Lacazette as he looked to pounce.
Wenger's men were rampant and only a block from Adam Smith, seemingly off his arm, kept out a Kolasinac volley after a smooth Arsenal attack.
Bournemouth started the second half with more intent as Jermain Defoe headed against a post, only for Arsenal to go down the other end and wrap up the points.
Lacazette robbed Bournemouth of possession and the ball broke for Aaron Ramsey who rolled a pass into Welbeck with the England international tucking home his second with a low finish.
The former Manchester United man should have had his hat-trick on the hour mark but his attempted chip dropped wide of Begovic's goal.
Welbeck was replaced with 15 minutes remaining as Sanchez was introduced to a mixed reception - the Chilean was soon into the action as he set up fellow substitute Olivier Giroud to again test Begovic.
Arsenal saw the game out with relative ease, Petr Cech saving a late Joshua King header aside, to pick up their second win of the season having been on top for much of the contest.