Arsenal went ahead on seven minutes when Samir Nasri passed to Cesc Fabregas on the edge of the area and neither Kevin McDonald nor Clarke Carlisle could get it off him as he weaved through and fired into the bottom corner.
Andrey Arshavin then struck the foot of the post from 18 yards.
The Clarets levelled on 28 minutes when Andre Bikey was felled by a late Thomas Vermaelen lunge and Alexander stepped up to fire home.
Arsenal suffered a blow as Fabregas was forced off two minutes before the interval but almost reclaimed the lead as Vermaelen met a Nasri cross with a powerful header and Jensen saved.
In the second half Burnley’s Chris Eagles blasted against the post as the game ended all square.