Alexandre Lacazette added a deserved second before half time, and Welbeck then doubled his personal tally early in the second half to cap off the scoring.
Find out how all of the action unfolded courtesy of our minute-by-minute coverage below.
Neither of these two teams have made a good start to the season, with Arsenal suffering back-to-back defeats before the international break and Bournemouth still looking for their first point of the campaign.
Both will be looking to bounce back today following the international break, but before we have a closer look at the situation for both teams let's first check out the team news...
Well, the headline news is that Alexis Sanchez only makes the bench for Arsenal having failed to secure a move to Manchester City on deadline day. There were some doubts over the Chilean's fitness following his time away during the international break, which saw Chile lose both of their games, but his absence from the starting lineup today is still bound to cause a stir.
Sanchez only made his first start of the season last time out against Liverpool, but today he loses his place in the side to Alexandre Lacazette.
The midfielder takes the place of Surman, who drops to the bench alongside the likes of Ibe, Pugh and captain Simon Francis, the latter of whom has failed to overcome a hamstring injury in time to start this one.
Defoe is, of course, a former Tottenham Hotspur player and has scored against Arsenal for three different clubs in the Premier League - Spurs, West Ham and Sunderland.
The only player to have scored against Arsenal for more clubs is Darren Bent, who has netted for five separate team against the Gunners.
Gosling and Arter, meanwhile, will patrol the middle of the park.
Rarely have the calls for Arsene Wenger to step down been so vociferous as after that defeat - which was more flattering for Arsenal than it was for Liverpool - and the Frenchman could have a long season ahead if things don't improve significantly.
On that occasion they only picked up one point from the first 12 on offer, eventually finishing the campaign 10th in the table.
In addition to the defeats to Stoke and Liverpool before the international break, there was also disappointment for Arsene Wenger's side in the transfer window as they missed out on Thomas Lemar - despite tabling a record £92 bid on deadline day. The collapse of that deal did mean that Sanchez stayed at the club, but the Chilean is out of contract next summer and the Gunners now look to have missed their chance to really cash in on him.
The Cherries are winless and pointless from their three Premier League games so far, leaving them in the relegation zone at this early stage of the season. There is still a long way to go, of course - and Bournemouth have lost their opening two games in both of their previous Premier League campaigns before recovering well - but Eddie Howe will wants those first points on the board as quickly as possible.
Bournemouth have a difficult run of fixtures ahead of them - including the Man City game they face five of last season's top seven in the space of eight games - so those losses to the Baggies and Watford could prove to be even more damaging over the coming months.
The Cherries brought Nathan Ake back to the club from Chelsea in addition to signing a proven Premier League goalscorer in Defoe and goalkeeper in Begovic. All three have quickly become important parts of the team, and on paper certainly look to have strengthened the Bournemouth squad.
Both sides are in desperate need of a win from this one, but Arsenal especially will be demanding a response from that Liverpool rout. The home advantage could be important, despite what may be a hostile environment if things don't go the hosts' way, and the Gunners should have enough to claims all three points here.
Sports Mole says: Arsenal 2-0 Bournemouth
That solitary draw did come in the most recent meeting, but for Bournemouth it would have felt like another defeat as they let a three-goal lead slip in the final 20 minutes. Daniels, WIlson and Fraser had put Bournemouth 3-0 up before the hour mark, but Arsenal stormed back late on with goals from Sanchez, Perez and Giroud.
The Cherries have faced Arsenal away from home on three previous occasions, and each one has ended in defeat with just one goal scored and eight conceded in that time.
ARSENAL STARTING XI: Cech; Mustafi, Koscielny, Monreal; Bellerin, Ramsey, Xhaka, Kolasinac; Ozil, Lacazette, Welbeck
BOURNEMOUTH STARTING XI: Begovic; Cook, Ake, Mings; A Smith, Gosling, Arter, Daniels; King, Defoe, Fraser
Ramsey plays the ball down the left wing for Kolasinac to race onto, and he stands a cross up in the middle having streaked clear of his marker. The cross is too high for Lacazette, but Welbeck is waiting behind him to steer his head into the corner and past a wrong-footed Begovic.
The hosts now have a free kick in a very dangerous position...
7 - Since August 2016, only Marcelo (11) has provided more assists among defenders in Europe's big five leagues than Sead Kolasinac (7). Up.— OptaJoe (@OptaJoe) September 9, 2017
Arsenal continue to boss possession here, keeping the pressure on Bournemouth, who have been unable to get into any sort of rhythm in this match so far.
Arsenal double their lead just before the half-hour mark, and it is a superb finish from Lacazette as he makes it two goals in as many home Premier League games. It is avoidable from Bournemouth's point of view as Ake had the chance to intercept the pass, but after he missed the ball Welbeck laid it off for Lacazette, but curls it into the top corner from around 25 yards.
It is a long way back for Bournemouth already!
2 - Alexandre Lacazette is only the 2nd Arsenal player to score in his first two home @premierleague games, after Marouane Chamakh. Poise. pic.twitter.com/tcEeHVPhdx— OptaJoe (@OptaJoe) September 9, 2017
Welbeck was quick to close the keeper down and his attempted clearance hits the Arsenal striker before falling to Lacazette inside the box. The striker really should score, but Begovic redeems himself by thwarting Lacazette and making a big save - even popping the ball in the process!
Chelsea are also ahead against Leicester thanks to an Alvaro Morata strike, while Watford have taken the lead against Southampton through Doucoure.
They lead Bournemouth 2-0 at the interval, and it is a fully deserved lead too. Arsenal have dominated from start to finish and, while there haven't been a huge number of clear chances, they have kept Bournemouth under pressure throughout.
Kolasinac's cross was met by Welbeck, and while he didn't make perfect contact with his header, he did get enough on it to steer it past Begovic. It looked as though the final touch came off his shoulder, which if anything helped to take it past the keeper.
Better from Bournemouth, though!
It is a second for Danny Welbeck as he slots his finish across goal and into the bottom corner after being played through by Aaron Ramsey. Again it is avoidable for Bournemouth, though, with Gosling having been caught in possession inside his own half at the beginning of the move.
The striker skips past a couple of challenges to burst into the box and then tries an audacious chip over Begovic which drifts inches past the far post. Brilliant from the England international, who deserved a goal there.
Arsenal come up with a convincing response to that Liverpool defeat with a 3-0 victory over Bournemouth, and it could have been more for the Gunners. They dominated from start to finish, with a brace from Welbeck either side of a Lacazette strike handing them a comfortable victory over Eddie Howe's side. The Gunners needed a response following back-to-back defeats, and they provided one today.