The Clarets at least ended a run that had seen them beaten in their previous 11 games against the Gunners, but the home side should have had more.
After riding some early pressure from the visitors, Burnley gradually asserted themselves to bombard the Arsenal goal with headed efforts.
Jeff Hendrick and James Tarkowski were most culpable of missing with headers, both failing to hit the target from close range, while Jay Rodriguez volleyed against the underside of the bar when he should have scored.
Arsenal extend their unbeaten run to seven games in all competitions, as well as being unbeaten in seven successive Premier League away games.
In truth, though, they were uninspired for long periods of this game, failing to gain a boost from the return of captain Pierre-Emerick Aubameyang after a three-match ban.
It was Aubameyang who had their best chances, miscuing when put through by David Luiz and later heading wide.
The draw sees both teams make marginal gains up the table, Arsenal to 10th and Burnley 11th, both on 31 points.