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A second-half double by Pierre-Emerick Aubameyang helped Arsenal finish the season with a victory and gave him a share of the Golden Boot.
Arsenal had faint hopes of overhauling Tottenham Hotspur in fourth place, but needed an eight-goal swing as well as a Spurs defeat by Everton.
After a goalless first half, Aubameyang claimed his 21st and 22nd PL goals of the campaign soon after half-time. The first came as he pounced on a defensive error on 52 minutes.
The second arrived 11 minutes later, Alex Iwobi providing the assist as Aubameyang's right-foot finish drew him level with Liverpool's Mohamed Salah and Sadio Mane on 22 PL goals this season.
Ashley Barnes headed a goal back for Burnley with 25 minutes left, but Eddie Nketiah's strike in stoppage time made it safe for Arsenal.
The Gunners finish fifth on 70 points, a point behind Spurs. Burnley are 15th, on 40 points.
See: Burnley report | Arsenal report
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