The former Real Madrid player, Arsenal’s captain, opened the scoring after five minutes following good work by Gabriel Jesus and doubled their lead from Ben White’s pull back.
Bournemouth failed to have a shot or touch in Arsenal’s penalty area in a one-sided first half and they were only marginally better in the second period.
Arsenal are top with nine points, two more than north London rivals Tottenham Hotspur, although champions Manchester City can dislodge them with a win at Newcastle United tomorrow.
Bournemouth have suffered back-to-back defeats after beating Aston Villa on the opening day of the season.
Harry Kane set a Premier League record as his headed goal earned Tottenham Hotspur a 1-0 home victory over Wolverhampton Wanderers.
Tottenham were outplayed in the first half but were a different proposition after the interval with Kane pouncing in the 64th minute to score his 185th Premier League goal.
No other player has scored more Premier League goals for a single club, with England skipper Kane moving clear of Sergio Aguero’s tally for Manchester City.
It was Tottenham’s 1,000th Premier League goal at home and lifted them to seven points from their opening three games.
Wolves have one point from their opening three games.
Fulham sealed their first victory of the Premier League campaign by beating Brentford 3-2 thanks to a late Aleksandar Mitrovic winner, after Bobby De Cordova-Reid scored the fastest goal of the season and Joao Palhinha produced a bullet header.
De Cordova-Reid’s goal came after just 44 seconds.
Two goals from Wilfried Zaha and a late strike by Jean-Philippe Mateta helped Crystal Palace come from behind and defeat Aston Villa 3-1 at Selhurst Park for their first win.
Palace are 10th with four points, one point ahead of Villa.
Southampton forward Che Adams scored twice after coming off the bench as the south-coast club came from behind to beat Leicester City 2-1 at the King Power Stadium.
Southampton are up to 12th on four points, while Leicester are 18th without a win and just one point.
Winger Demarai Gray fired a late equaliser to give Everton a 1-1 draw with Nottingham Forest at Goodison Park as they picked up their first point of the new Premier League season following a rocky start.
Forest stay 11th on four points and Everton are up to 16th with one point.
In the Sunday games Manchester City are at Newcastle, Leeds host Chelsea and West Ham host Brighton.
On Monday (UK time) Manchester United play Liverpool and Old Trafford.