Battling Burnley kept Arsenal at bay for over an hour before two quickfire goals finally broke their resistance.
The Clarets faced a first half barrage from the Gunners and had Tom Heaton to thank for two great saves from Alexis Sanchez.
But the Chilean, who cost a reported £30m, finally got his revenge with a firm header 21 minutes from time.
And within 60 seconds Calum Chambers added a second to further ease the growing tension inside the Emirates.
Man of the match Sanchez then wrapped up the points in stoppage time with a class finish.
Burnley were restricted to half chances throughout, with Danny Ings a whisker away from adding to last week's goal early in the second half.
The Clarets made two changes, with Dean Marney returning to midfield after a four-match absence with a hamstring injury, and Marvin Sordell preferred in attack to Lukas Jutkiewicz.
And it was a tight start by the Clarets, who operated with Ings sitting just off Sordell.
Ings and George Boyd we're heavily involved as the visitors broke after nine minutes, but finally Arsenal clears their lines.
But the patient Arsenal build up play almost paid dividends soon after when England striker Danny Welbeck played a quick one-two and danced across the box, only for a retreating Kieran Trippier to block his goalbound effort.
There were Arsenal appeals for a penalty midway through the half when Boyd, booked earlier for a foul on Santi Cazorla, inexplicably raised an arm and made contact from a right wing corner.
However, referee Craig Pawson and his assistants gave the midfielder the benefit of the doubt, despite the vehement appeals of Per Mertesacker.
Back came Arsenal and following a sensational turn from Sanchez and cross from Chambers, Cazorla rattled a volley inches wide of Tom Heaton's left hand post.
But Burnley were not content with defending and just after the half hour, Scott Arfield was close to teeing up Ings on the penalty spot following good work by Boyd.
And following a smart save by Heaton, diving to his left to claw away Sanchez's 18 yard curler, Ings was just off target from distance.
The Gunners were now all guns blazing though and Sanchez, instrumental in all the hosts good work, shaved the upright with another sensational right-foot strike after cutting in from the left.
The Chilean was a constant threat - and after hesitancy in defence allowed Cazorla to pick him out in the left channel with a slide-rule pass three minutes from the sanctuary of the break, Heaton again came to the rescue with a smart block.
The restart saw Burnley tear into Arsene Wenger's side, with Ings a stud away from sliding the ball past Wojciech Szczesny.
And in the opening 10 minutes of the half, it was the visitors forcing mistakes from Arsenal as the Emirates crowd grew ever more frustrated.
Wenger's response was to replace captain Mikel Arteta with talisman Aaron Ramsey.
And with a quarter of the game remaining Arsenal thought their moment had come.
Mertesacker picked out Chambers with a ball in behind Stephen Ward and when the ball was cut back into the six yard box, Cazorla looked certain to score only for Michael Duff to make a miraculous, goal-saving block at full stretch.
Arsenal could not believe it, but Burnley were finally undone with 20 minutes remaining.
Chambers picked out dangerman Sanchez with a peach of a cross and the striker's downward header rom eight yards beat Heaton.
And within two minutes the game was up for Burnley as Chambers took advantage of a corner that wasn't cleared to drill home from close range.
Heaton's terrific, instinctive save on the line denied Lukas Podolski.
And the Clarets keeper then capped an impressive display by denying sub Theo Walcott and Ramsey before Podolski almost uprooted the post with a stunning volley.
But Sanchez had the final say, slotting home from close range in the final seconds to cap a fabulous individual display.
Burnley: Heaton, Trippier, Duff, Shackell (c), Ward; Arfield, Marney (Chalobah 80), Jones, Boyd; Ings, Sordell (Jutkiewicz 68)
Burnley unused subs: Gilks, Keane, Kightly, Wallace, Mee
Arsenal: Szczesny; Chambers, Mertesacker, Monreal, Gibbs; Cazorla, Arteta (c) (Ramsey 64), Flamini, Sanchez, Oxlade-Chamberlain (Walcott 80), Welbeck (Podolski 80)
Arsenal unused subs: Rosicky, Sanogo, Marinez, Bellerin