Arsenal were held to a draw at Turf Moor after a disastrous error from Granit Xhaka allowed Chris Wood to cancel out Pierre-Emerick Aubameyang's early opener.
Both sides had chances to win the game, but the visitors were left the more disappointed, unable to secure another win despite a couple of near-penalty incidents and good opportunities late on.
Here are five things we learned from the game.
Reinvigorated Aubameyang key to Arsenal form
Arsenal have, for several years, been heavily reliant on the prolific goal scoring of Aubameyang. He has repeatedly turned draws into wins, produced something out of nothing when his side most needed it.
But this season he has been oddly subdued and dipped well below his usual rate of scoring. It's no surprise, then, that Arsenal were languishing in the bottom half of the Premier League table for so long and now sit in midtable, a position that doesn't feel in any way false.
If the Gunners are to push for a European place - most likely a Europa League spot - Aubameyang's form is vital. And his upturn of late has coincided with a general improvement from Arsenal. The relentless consistency isn't quite there yet, but a hat-trick against Leeds and a brace against Benfica, followed by a well-taken goal at Burnley, suggest he is returning to his previous levels.
The goal here was vintage Aubameyang, too: a shift inside from the left and a powerful precise finish at the near post (though Nick Pope might feel he could have done better).
Turf Moor a tough place to go?
There has been an enduring myth that playing at Turf Moor is a test of steel and determination for even the most gifted teams, that leaving with a victory always requires a scrap and a battle with the elements. That is probably because it is often windy, and because Sean Dyche's team are the embodiment of English football's traditional grit and ruggedness. Like away games at Stoke in years gone by, the idea is largely romanticised and doesn't really have any basis in fact.
Burnley had picked up 17 points from 14 home games before the visit of Arsenal, hardly the most eye-catching record. They were beaten convincingly by Manchester City and Chelsea here, lost to Bournemouth in the FA Cup and were held to a tame draw by struggling West Brom.
But there have been some signs that Dyche's side are beginning to put a run together on home turf, even if they have not won at this stadium since late January. The draw against Leicester was encouraging, and a result against a confident Arsenal side should provide something to build on.
Willian finally finding his feet
There are unlikely to be thousands of Arsenal fans hailing Willian's arrival this summer as the coup of the century because of a string of improved performances, but there might at least be some begrudging respect now. Plenty have written off the Brazilian, now 32 and apparently struggling to settle into any consistent form at his new club.
In recent weeks, though, Willian appears to have found his feet. He was the creator of Aubameyang's goal at Turf Moor, driving through the middle before sweeping a pass out to the left-sided forward, who did the rest. That makes it three assists in three appearances for the former Chelsea man, ending a run of 13 appearances without one.
The logic of handing Willian a three-year contract will still, rightly, be questioned. But he is beginning to remind us all of his ability to influence Premier League games. How long this will last remains to be seen.
Errors continue to plague Arsenal
As was the case at Wolves last month, Arsenal let themselves down here with an almost inexplicable mistake. They had dominated the first-half, controlled the game and kept Burnley at arm's length, but Granit Xhaka's lapse in concentration gifted the hosts an equaliser just before half-time.
It was entirely unavoidable, too: Bernd Leno played a pass to the midfielder, inside his own area, and needlessly put Arsenal in an awkward position. Matej Vydra applied pressure, forcing Xhaka to play a rushed pass towards Leno. But he succeeded only in firing the ball against Chris Wood's hip. The ricochet flew past Leno and into the net, and in an instant the visitors' good work throughout the opening 45 minutes was undone.
This is the kind of thing Mikel Arteta will be desperate to coach out of his players. This just doesn't happen to Manchester City, whose players, by now, seem hardwired to play out from the back with maximum efficiency. If Arsenal want to challenge at the top of the Premier League again, these unforced errors need to be eradicated.
More handball controversy
Does anyone really know what's going on anymore? The players don't seem to. Nor do the pundits, or the fans. Erik Pieters, having been introduced as a second-half substitute, was twice involved in potential handball incidents in his own box. The first looked a more viable penalty shout as Nicolas Pepe struck the ball against the defender's arm, but nothing was given after a VAR check.
The second was more dramatic. A goal-bound shot from Pepe was blocked by Pieters, seemingly with his right arm. The referee pointed to the spot and the Dutchman was sent off. But that decision was soon reversed after it became clear the ball had struck his shoulder.
Regardless of the outcome in this instance, the needlessly complex interpretation of the rule is becoming a problem.