The Gunners needed a minor miracle to secure Champions League qualification via the league but missed out to Tottenham, despite a first domestic win in five matches.
Burnley end the season with three successive defeats but given the position they found themselves in at Christmas time, staying in the top flight is a major achievement for Sean Dyche's side.
In an open start to the game, Aubameyang should have opened the scoring after just three minutes when he got the wrong side of Ashley Barnes, only for the ball to hit his shoulder and bounce onto the post.
Barnes then wasted two glorious chances, flashing his header wide and then driving straight at Bernd Leno.
As Burnley gained control of the match, they passed up chance after chance to open the scoring, Chris Wood hitting the base of the post when one-on-one with Leno.
Shkodran Mustafi then had to be on his toes to block Matthew Lowton’s goal-bound effort while Leno was on the ground.
Having failed to make the breakthrough, Arsenal nearly made the home side pay in first-half stoppage time, Joseph Willock turning on the edge of the box and nearly finding the bottom corner with his strike.
It was a different story after the interval, a change of formation from Unai Emery paying dividends as Aubameyang opened the scoring after 52 minutes when Jack Cork’s misplaced pass fell to the Gabon international, who made no mistake when one-on-one with Tom Heaton to score his 30th of the season.
Aubameyang grabbed a second just after the hour mark, volleying past Heaton when found at the far post by Alex Iwobi. That goal was enough to secure the Gunners frontman a share of the Premier League golden boot, tied on 22 goals with Liverpool duo Sadio Mane and Mohalaned Salah.
Burnley responded quickly to falling two behind, Ashley Barnes pulling a goal back two minutes later when steering home Johan Berg Gudmundsson's cross at the near post.
Aubameyang then missed a great chance for his hat-trick when failing to score from four yards before substitute Peter Crouch, potentially making his last appearance for the Clarets, saw his goal-bound effort blocked by James Tarkowski.
As Burnley faded, Arsenal grabbed a third in stoppage time, Eddie Nketiah netting his first Premier League goal as he squeezed the ball through Heaton’s legs.