Arsenal dominated proceedings but struggled to break down a resilient Burnley defence in the first-half.
The two sides went in level at the break but the Gunners finally made the breakthrough in the second-half on 59 minutes.
Shkodran Mustafi was the unlikely scorer, heading home a Mesut-Ozil corner for his first Arsenal goal.
The floodgates looked set to open for the home side as Burnley attempted to push for an equaliser.
However, Granit Xhaka made for a shaky last half-an-hour for the Gunners after being sent off.
Referee Jon Moss showed the Swiss international a straight red card for a sliding challenge on Steven Defour.
And Moss caused further controversy when he awarded Burnley a late penalty following a Francis Coquelin challenge on Ashley Barnes.
Andre Gray stepped up to take the spot-kick and duly converted but the drama was not done there.
In the seventh minute of added time Moss once again pointed to the penalty spot after Ben Mee appeared to kick Laurent Kosccielny in the face.
And Alexis Sanchez made no mistake from the spot, calmly dinking it down the middle past Tom Heaton.
The result sees Arsene Wenger's side register back-to-back wins following last week's 4-0 victory over Swansea.
The Gunners leap-frog Liverpool and Tottenham into second place in the Premier League.