By Chris Harris at Turf Moor
Aaron Ramsey marked his 150th Arsenal appearance with the only goal as the Gunners dug deep to win at Burnley.
The Wales midfielder found the roof of the net in the 11th minute after Mesut Ozil's shot was saved and Alexis' follow-up ricocheted off two defenders.
It was a scrappy goal but, on Grand National Day, it was enough for the Premier League's form horse to make it eight wins in a row and ensure another week in second place.
Arsenal had other chances - Ozil and Santi Cazorla going close in the first half - but Burnley stretched them to the limit and David Ospina was at his best to deny Sam Vokes and Kieran Trippier.
If last weekend's stellar victory over Liverpool owed much to the flair of Arsene Wenger's team, this triumph was forged on defensive discipline.
How encouraging that Arsenal can call upon different qualities to keep on picking up points.
Now for Wembley - and an FA Cup semi-final.
SETTING THE SCENE
After 15 wins in 17 games, it was no surprise to Arsene Wenger name an unchanged line-up for this trip to a blustery Turf Moor.
Laurent Koscielny shrugged off a thigh problem to take his place alongside Per Mertesacker, but fellow centre back Gabriel missed out through illness. Wojciech Szczesny returned to the bench after a hip injury and Calum Chambers was also back in the squad.
There were two notable milestones aside from Ramsey's. Cazorla completed a century of Premier League appearances and Olivier Giroud prepared for his 100th Arsenal start.
The Gunners' improving form against the top flight's leading lights has been well-documented, but Burnley could say the same. Sean Dyche's side had beaten Manchester City and drawn with Chelsea and Tottenham in recent times, and their home form was their trump card in the fight against relegation.
Arsenal started brightly and fashioned a chance within seconds.
Alexis, his belief restored by a stunning goal against Liverpool, earned a free-kick 25 yards out and took it himself. The Chilean applied plenty of power and goalkeeper Tom Heaton was forced to make a late adjustment as the ball moved in the air.
The visitors oozed confidence in the early stages, so much so that Giroud tried to find the net with a 35-yard volley.
But Burnley were no shrinking violets, and Ospina had to make a vital early block after Vokes outpaced Mertesacker in the left channel and shot from just inside the box.
Alexis got underneath a half-chance after a loose ball fell kindly to him, and the same man was involved in the move that gave Arsenal the lead.
Francis Coquelin's interception turned defence into attack in an instant, Alexis' initial shot was deflected to Ozil, Heaton kept out the German's fierce effort, Alexis' follow-up bounced off two defenders and Ramsey fired the loose ball into the roof of the net.
It wasn't the prettiest goal but it was one that Arsenal deserved. They almost added to their lead when Cazorla curled a delicious free-kick just beyond the top corner of Heaton's goal.
Burnley went close with a free-kick too, Trippier bringing the best out of Ospina from 25 yards towards the end of an entertaining half.
The home side began the second half on the front foot, urged on by a crowd that felt the referee had given them nothing.
Sturdy defending from Nacho Monreal and then another save from Ospina kept Burnley at bay, but their pressing game regularly forced Arsenal into rushed passes and hastily-made clearances. You could see how big names have been humbled at Turf Moor.
Arsenal remained a threat - Alexis testing Heaton with a low shot from distance - but they were playing on the break more often than not.
Ozil set up a half-chance for Ramsey which he opted to take down instead of volley, giving a Burnley defender time to block. And Cazorla arrowed a long-range effort just over and the visitors alleviated the pressure.
Back came Burnley. Vokes turned brilliantly inside the Arsenal box and Monreal had to stay calm to watch the ball, dispossess the striker and clear the danger.
Moments later, Ben Mee got away from Bellerin and clipped the ball to the far post, where George Boyd just failed to connect. Then Danny Ings, a constant menace to Mertesacker and Koscielny, had an overhead kick saved by Ospina.
Arsenal needed some sustained possession to take the sting out of the game and for a while they got it, without creating anything of note - although Cazorla tried his luck again from outside the box.
Giroud was replaced by Danny Welbeck and he almost profited from a slick counter-attack, but having been forced wide by Ozil's pass he dragged his shot across goal.
It didn't matter. Arsenal had done enough - just.
Referee: Mike Dean