On a frantic final day, Arsenal finished below north London rivals Tottenham Hotspur, who beat Leicester City 4-2 to seal the seventh place.
Arsenal dominated possession in the first half and went close to opening the scoring but Rob Holding's goal from a set-piece was ruled out for offside while his centre-back partner Gabriel saw his floating header come off the crossbar.
Pepe gave Arsenal the lead four minutes after the restart, controlling a low cross from Calum Chambers with his left foot before firing a fierce strike with his right foot past Robert Sanchez in Brighton's goal.
Arsenal doubled the advantage when Martin Odegaard released Pepe down the right channel and the Ivorian forward composed himself to curl a low finish inside the far post for his 10th league goal of the season.
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