England became the first country to provide two finalists in both European competitions in the same season after Chelsea and Arsenal made it through to the title match in the Europa League on Thursday.
The win kept Arsenal in fourth place, while Bournemouth remained in 12th.
Mesut Ozil, who was born in Germany to Turkish-origin parents and lifted the World Cup in 2014.
Pierre-Emerick Aubameyang struck twice as Arsenal defeated Ukraine's Vorskla Poltava 4-2 in the Europa League. Danny Welbeck and Mesut Ozil were also on target before Vorskla hit back with late goals.
IN PICTURES | Arsenal lose tough second-leg game against Ostersunds, scrape through to Europa League last 16
Two quickfire first-half goals gave Swedish minnows Ostersunds hope, but Sead Kolasinac's goal just after the interval saw Arsenal hang on for a 4-2 win on aggregate.
Cup holders Arsenal were sensationally dumped out of the competition by Nottingham Forest, joining Leeds United as big-name casualties on a day of shocks.
Wenger has been the manager of Arsenal since October 1996, where he has since become the club's longest-serving manager and most successful in terms of major titles won.
Aaron Ramsey and Olivier Giroud scored as Arsenal twice came from behind to beat Leicester City 4-3 in a dizzily entertaining opening Premier League game.
Arsenal collected the Community Shield after beating Chelsea 4-1 in a penalty shootout that used an experimental format mixing up the order of the spotkicks.