Unai Emery’s side moved above Leicester City into third on 15 points, nine behind leaders Liverpool after eight games.
Arsenal moved up to third in the Premier League standings after David Luiz scored his first goal for the club in a nervy 1-0 win at home to Bournemouth on Sunday in which the visitors missed a flurry of chances to level in the second half.
The former Chelsea defender darted across the area undetected to glance a Nicolas Pepe corner into the far corner to give Arsenal the lead in the ninth minute.
Arsenal's record signing Pepe came close to doubling their lead with a shot that just missed the top corner while Pierre-Emerick Aubameyang also saw a strike from distance flash just wide in the first half.
Bournemouth, who were beaten 5-1 on their last trip to the Emirates Stadium, produced a cagey display in the first period although they nearly equalised when Dominic Solanke narrowly missed the target with a header from a free kick.
They showed far more ambition in the second half while Arsenal lost focus but failed to take their chances, with Callum Wilson mis-placing a pass in the area after rounding Bernd Leno and later seeing a shot tipped over by the German goalkeeper.
Wilson missed two further chances in the closing stages. Steve Cook headed over from a corner as Arsenal somehow clung on for the victory, also recording their first clean sheet since beating Newcastle United 1-0 on the opening day of the season.
Unai Emery's side moved above Leicester City into third on 15 points, nine behind leaders Liverpool after eight games.
The supporters sang "1-0 to the Arsenal" as fulltime approached in a nod to the scoreline which they were renowned in the George Graham years but there was little that was convincing about their defence here, even though they held on for the win.
Arsenal had shipped five goals in their previous three league home games and Emery said he had spoken to his players of the importance of taking control of the game.
"We said we wanted to take a step ahead in this match after the three games. We controlled the first half, we were organised and we got close to their area a lot. The first half is a good example of how we should play," he told a news conference.
"I'm critical of the team but also very positive. We know our obligation is to win but how we win is also very important."
Bournemouth are 10th on 11 points and coach Eddie Howe criticised his side for their first-half display but said they deserved more for their efforts after the break.
"We were a little bit too passive and didn't dominate our style on them. But you saw the other side in the second half," he said.
"We dominated possession and deserved a point, we did well enough and had good enough chances. When we play well we're a match for anybody, if we'd played in the first half like we did in the second we'd have won the game."