The Gunners' victory was assisted by the controversial sending-off of Burnley's Northern Ireland international Kyle Lafferty for a challenge on Gilberto Silva.
It was Lafferty who almost got the Clarets off to the perfect start with a fifth minute header from an Andy Gray cross which cannoned back off the crossbar.
The home side were made to pay for their lack of luck in front of goal when just three minutes' later Kolo Toure chipped a pass over the static Burnley defence for Eduardo to virtually walk the ball past home goalkeeper Gabor Kiraly.
The home team's response was positive with striker Gray forcing a save from Arsenal goalkeeper Jens Lehmann. Burnley then created a great opportunity as Graham Alexander's cross was blocked by the Arsenal defence as far as Robbie Blake, but he fired over the crossbar.
At the other end Owen Coyle's side were indebted to midfielder James O'Connor, who was in the right place at the right time to head off the line, after Gilberto had met a right-wing corner with a strong header.
The Gunners had another escape eight minutes before the half-time interval when left-back Armand Traore was caught in possession at the far post by Wade Elliott who laid the ball back for Chris McCann to fire over the top.
Striker Nicklas Bendtner sprang into life with an acrobatic overhead kick five minutes before the break, which forced a good save out of Kiraly.
Five minutes into the second half Eduardo should have made the game safe as he played a one-two with Bendtner but placed his shot wide of the target.
Five minutes later McCann missed another opportunity, this time heading over from a Blake free-kick.
Burnley's task was made harder on the hour mark as Lafferty was shown the red card for a strong challenge on Brazilian Gilberto.
Arsene Wenger's side didn't need a second invitation to make the most of the extra man and killed off the plucky Championship side 16 minutes from time, when Bendtner latched on to a through ball from Eduardo to take the ball around Kiraly and score with ease.