Goals from Cesc Fabregas, Theo Walcott and Andrey Arshavin secured the points but the Gunners made it a lot harder for themselves than it should have been.
Burnley's goal came from David Nugent and that lone effort was enough to generate nervousness amongst home fans until almost the very last kick.
The Gunners began brightly and had two good chances to take the lead inside the first four minutes.
The first came when Nicklas Bendtner was put through by Fabregas on the right side of the area. The Gunners striker never quite got the ball under control and his tame shot was palmed away by Brian Jensen .
The resulting corner was delivered by Fabregas and eventually fell to Thomas Rosicky, whose first-time shot was just inches wide with Jensen a spectator.
Walcott looked like he had a real appetite for the game and was a thorn in the side of both Daniel Fox and Chris Eagles, who were doubling up on the England winger.
Some good interplay between Fabregas and Emmanuel Eboue, including a clever back-flick by the Ivorian, allowed the Gunners captain free run on goal.
Unfortunately, his shot was dragged wide across goal and bobbled clear of the right upright for a goalkick.
Arsene Wenger's side went ahead in the 34th minute through Fabregas, who got onto the end of Samir Nasri's loping ball over the top of the Burnley rearguard to half-volley past Jensen from four yards out.
That was Fabregas' last real contribution of the game as he was replaced by Abou Diaby, who had been warming up in readiness for some 10 minutes prior to the goal.
Bendtner spurned two golden chances within the first two minutes of the second half and the Gunners were regretting the misses when Burnley equalised on 50 minutes.
Leon Cort headed a long ball into the Arsenal area and Nugent sprung the offside trap and then lifted the ball past a helpless Manuel Almunia.
After that, Burnley pulled everyone back in defence while the Gunners continued to carve out opportunities but found a goalkeeper in form in Jensen who made magnificent saves from Walcott, Samir Nasri and Bendtner.
The big shot-stopper was eventually beaten by a delightful run and shot by Walcott on the hour to cap a performance where he had been Arsenal's most dangerous player.
More chances went begging as the clock ticked away. Arshavin missed a gilt-edged chance when teed up by Diaby, who had broken away from the Burnley defenders with seven minutes to go.
Jensen denied Walcott with a fine save a minute later and a crucial third goal still would not come.
Massive relief came for the visitors in the last minute of added time when Arshavin suddenly broke from the right to accelerate into the box and fire across goal and into the bottom left corner of the net as Arsenal's unlikely late charge for the title gained a little more momentum.