Perhaps it’s just not meant to be as the Clarets were left waiting for their first Premier League win over Arsenal and a first in the top tier since 1974.
But after a terrific Turf Moor performance - and a first home draw in exactly a year - they could not have come closer.
And after a run of 10 successive league defeats against the former champions, stopping the rot was the least Sean Dyche’s men deserved as they made it seven points from three games to go into the mid-winter break in great shape.
The hat-trick of wins after the victories over Leicester and Manchester United that had transformed the Clarets’ fortunes proved cruelly elusive as Jay Rodriguez went closest to a late winner with an effort that bounced down off the bar and away via the goalline.
But Burnley could have given no more from a tremendous collective effort which earned tired limbs a deserved few days off.
And to remain level on points with Arsenal and 11th in the table leaves spirits high for the final push ahead.
Not since their first visit to Turf Moor in the Premier League have the Gunners dropped points but they rarely get an easy ride.
And that was certainly the case again once the Clarets – unchanged from their history-making win at OId Trafford – had survived an anxious start.
Pierre-Emerick Aubameyang – so often a thorn in Burnley’s side – was back from a ban for the visitors and missed the kind of chance after 14 minutes he invariably buries in the fixture.
Released by David Luiz’s ball over the top the Arsenal captain dragged his effort wide and then when getting into a similar position later in the half failed to beat Nick Pope, who read the danger smartly to dash off his line and smother an attempted lob.
James Tarkowski, again playing like his life depended on it, also had to make a vital intervention to stop Alexandre Lacazette from pulling the trigger in front of goal.
But the Clarets were determined not to let Mikel Arteta’s side have it all their own way and as they wound up the pressure, the Londoners were the ones hanging on and glad to hear the half-time bell.
Rodriguez had forced Bernd Leno into an early save from a skidding effort that was bottom-corner bound and Dwight McNeil also fizzed a shot over the top after Matt Lowton – enjoying a terrific first – half had won the ball back in a style that had Burnley on the front foot.
Set-pieces were also a threat for the Clarets and seven minutes before the break Dyche’s men almost broke the deadlock.
Ben Mee’s pass released McNeil for a typical surge up the left and when he crossed low, Jeff Hendrick came off the opposite flank to side-foot an effort just wide of the target.
Arsenal were struggling to cope with the Clarets’ intensity and there was no let-up after the break as Burnley pressed even harder for the opening goal.
Luiz managed to block a Chris Wood header from a McNeil cross and when the young winger delivered again at the end of an excellent move Hendrick was just off target with another headed effort from the edge of the six-yard box.
The Gunners struggled to cope whenever Burnley whipped the ball into the box and from another Ashley Westwood free-kick, Rodriguez got just too much on a near-post flick that took the ball wide.
Arsenal remained a threat on the break, however, and with 15 minutes to go they almost grabbed the lead on the counter-attack when Lacazette crossed for Aubameyang to head inches wide.
No-one came closer, though, than Rodriguez to finding a winner as the Clarets were denied by the width of the goalline.
Westwood’s cross was met in the air by McNeil and from his knock-down Rodriguez connected with a side-foot volley than hit the underside of the bar before bouncing down and out to the anguish of the striker and the crowd who willed the ball over the line.
It would have been a famous double for Rodriguez after his spectacular strike at Manchester United.
But with the fine margins not favouring the Clarets this time and the outstanding McNeil, again showing his young England class, then volleying a Hendrick cross over the bar, Burnley had to settle for a first draw since September and the point which keeps them in good company looking upwards.
Burnley: Pope; Lowton, Tarkowski, Mee, Taylor; Hendrick, Westwood, Cork, McNeil; Wood, Rodriguez. Substitutes: Hart, Brady Pieters, Lennon, Bardsley, Vydra, Long.
Arsenal: Leno; Bellerin, Mustafi, Luiz, Saka (Torreira HT); Guendouzi, Xhaka; Ozil (Willock 64); Martinelli, Aubameyang, Lacazette (Nketiah 88). Substitutes: Martinez, Sokratis, Ceballos, Pepe.
Booked: Ozil, Xhaka, Torreira
Referee: Chris Kavanagh