The Gunners had been on course for a draw
Arsenal were on course for a draw at Burnley but a stoppage time goal gave the Gunners a narrow 1-0 victory.
A 92nd minute short corner from Mesut Ozil to Alexis Sanchez was headed towards the far post where Alex Oxlade-Chamberlain directed an effort towards goal that went in off Laurent Koscielny's arms.
The result is their fifth Premier League win in a row and lifts Arsene Wenger's men up to third, just two points leaders Manchester City who lost to second placed Tottenham in the earlier game on Sunday.
Arsenal dominated possession and the flow of the game but came up against a resolute Burnley rearguard.
The home side even almost took the lead when Michael Keane headed against the crossbar as Arsenal survived.