The Gunners needed both Tottenham and Everton to drop points while also seeing off the Seagulls to seal seventh place and a spot in the inaugural Europa Conference League next year.
Arsenal did their bit and had Pepe to thank, the winger's two second-half strikes securing a 2-0 victory as 10,000 fans returned to the Emirates Stadium.
But Gareth Bale's late goals for Spurs at Leicester meant they finish above their north London rivals.
With muted protests against the club's owners before the game and director Josh Kroenke watching on from the stands, this could have been a powder keg of emotions had Arsenal lost.
Instead, Mikel Arteta's side won comfortably – although it proved not to be enough in their quest to maintain a 25-year run of European qualification.
Martin Odegaard, who returns to Real Madrid after the game, showed why Arsenal are so keen to keep hold of the Norway international as he ran the show.
There we also strong performances from Thomas Partey and Granit Xhaka in the heart of the midfield.
Brighton were never really in the contest but did frustrate the home side for the opening 45 minutes, Partey and Xhaka coming close and Rob Holding bundling the ball over the line only to be flagged offside.
Arsenal came closest to breaking the deadlock just moments before the break when Gabriel Magalhaes sent a looping header against the crossbar.
But they would hit the front soon after the restart as Pepe continued a fine run of form, collecting a Calum Chambers pass and finishing coolly past Robert Sanchez.
The club's record signing would double the lead on the hour-mark, scoring his ninth goal from his last 13 Premier League starts having cut into the Brighton box and again finished with aplomb.
Partey would go on to strike the crossbar as Arsenal coasted to the three points to at least win in front of their returning supporters.
But the rebuild for Arteta and the club starts now in earnest with no silverware or European football at the end of a taxing season.
The majority of fans remained in place for a lap of honour from the Arsenal players but the interlude was filled with anti-Kroenke chants.