The hosts were already leading 2-0 thanks to strikes from Pierre-Emerick Aubameyang and Alexandre Lacazette when Sead Kolasinac decided to get in on the act.
Jack Wilshere did brilliantly, picking the ball up deep in midfield and driving forward to the edge of the Burnley box where he found Kolasinac overlapping down the left.
The defender got the ball out of his feet and rifled a low effort across Nick Pope an into the far corner to secure the points and ensure Arsenal claimed victory in Wenger's final game at the Emirates.