The Gunners started the game extremely quickly and had Bournemouth under pressure for the majority of the first half.
Mikel Arteta’s side had 11 shots in the opening 20 minutes of the match – the joint-most by a Premier League side this season – but for a while it looked as though it might not be their day.
William Saliba had a shot well saved by Travers after he exchanged passes with Saka, before the England winger himself had an effort saved by the Bournemouth keeper.
Rice had a good chance to open the scoring in the first half but he shot just wide from a Havertz knock down.
Trossard, Thomas Partey, Martin Odegaard and Takehiro Tomiyasu all had further opportunities before Saka finally made the breakthrough from the spot.
The tension eased when Trossard finished well to double the lead, and Rice’s strike in added time added further gloss to the scoreline.
The victory is Arsenal’s fourth win on the bounce as they keep the pressure on Manchester City and hope the champions drop points in the title race.