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  The Premier League side led when Calum Chambers superbly curled in from 18 yards with the outside of his foot.
  Championship side Burnley hit back as Sam Vokes headed in Tendayi Darikwa's cross and an Alex Iwobi interception denied Andre Gray after the break.
  Sanchez - making his first start since November - finished coolly to win it.
  The Chile international - who has made just one substitute appearance since suffering a hamstring injury in November - was lively throughout and turned in Alex Oxlade-Chamberlain's cross after Darikwa was dispossessed in midfield.
  There can be no doubt the holders deserved a place in Sunday's draw as they enjoyed the lion's share of possession and chances, though Burnley created a nervy finish.
  The impressive Gray caught sight of goal late on but could not get meaningful power in his shot and as the visitors threw bodies forward, substitute Theo Walcott missed a glorious chance to add a third on the break as he failed to round Tom Heaton.
  Arsenal's match programme featured the FA Cup trophy on the front cover along with the words "To have and to hold".
  They move a step closer to emulating the Blackburn Rovers side of 1886 by winning a third FA Cup in a row, though this was far from vintage.
  They made nine changes to the side beaten by Chelsea last weekend but still fired 23 shots at the Burnley goal as Arsene Wenger maintained his record of only losing to lower-league opposition once in 41 FA Cup ties.
  Solid appearances from the likes of Chambers and Iwobi show he has a squad capable of maintaining a challenge in this competition but the anxious moments - particularly with the ball in the air in their box - will continue prompt some concern.
  Mohamed Elneny saw more of the ball than any other player - with 100 touches - on his first outing since joining for £5m from Basel earlier this month.
  His completion of 96% of his passes was impressive but as a deep midfielder, he often found himself with plenty of time on the ball, meaning there will be tougher tests to come.
  "I felt he started a bit cautious, played a bit secure," said Wenger of Elneny. "He became more adventurous. It will take him some time to adjust to the power side of our game here, but the intelligence, the mobility and the technical level are good."
  While Elneny was steady between the penalty areas, Sanchez was pivotal inside the box.
  He did not break stride to slam home his goal, created the opener for Chambers and looked lively throughout. His return could be crucial ahead of a key February when Arsenal play the likes of leaders Leicester, Barcelona and Manchester United.
  Like Arsenal, Burnley sit third in their respective table and their offering at the Emirates shows they have the grit needed to win promotion back to the top tier at the first time of asking.
  Gray missed an early one-on-one chance and looked set to put his side 2-1 up after the re-start but for Iwobi's last-ditch intervention.
  His manager Sean Dyche was bold in making just two changes with a league fixture at Sheffield Wednesday to come on Tuesday.
  But the Clarets looked cohesive and more finishing of the quality Vokes displayed could have asked tough questions of the Premier League side.
  They undoubtedly look capable of staying the pace in the second tier.
  Arsenal manager Arsene Wenger: "Burnley are a good side. For the first 20 minutes we looked quite in control but when they equalised suddenly you could see, energy, commitment, and we had to dig deep to win the game. Every free-kick you worry because Sam Vokes is very strong in the air and he scored their goal - they are strong on set pieces and we needed to be alert."
  Burnley manager Sean Dyche: "It was a closer-run affair this year, and we had some good chances. I'm really pleased with the growth my players have shown in that year, but we didn't quite have enough and ultimately the disappointment is we haven't gone through."
  Arsenal host Southampton on Tuesday - where they could go top of the Premier League - and Burnley travel to Sheffield Wednesday on the same night as they bid to close a four-point gap to the automatic promotion places.
  Match ends, Arsenal 2, Burnley 1.
  Second Half ends, Arsenal 2, Burnley 1.
  Foul by Mikel Arteta (Arsenal).
  Joey Barton (Burnley) wins a free kick in the defensive half.
  Attempt missed. Mohamed Elneny (Arsenal) right footed shot from outside the box is close, but misses to the left.
  Olivier Giroud (Arsenal) wins a free kick in the defensive half.
  Foul by Fredrik Ulvestad (Burnley).
  Delay over. They are ready to continue.
  Delay in match Kieran Gibbs (Arsenal) because of an injury.
  Delay in match  (Burnley).
  Attempt missed. Andre Gray (Burnley) left footed shot from the left side of the box is close, but misses to the left. Assisted by Joey Barton.
  Attempt missed. Michael Kightly (Burnley) right footed shot from outside the box is too high. Assisted by Rouwen Hennings following a set piece situation.
  Foul by Olivier Giroud (Arsenal).
  Joey Barton (Burnley) wins a free kick in the defensive half.
  Mikel Arteta (Arsenal) wins a free kick in the defensive half.
  Foul by Ben Mee (Burnley).
  Offside, Arsenal. Alex Oxlade-Chamberlain tries a through ball, but Theo Walcott is caught offside.
  Delay over. They are ready to continue.
  Delay in match Stephen Ward (Burnley) because of an injury.
  Attempt missed. Alex Oxlade-Chamberlain (Arsenal) right footed shot from the centre of the box is too high. Assisted by Tomas Rosicky.
  Offside, Burnley. Stephen Ward tries a through ball, but Michael Kightly is caught offside.
  Mikel Arteta (Arsenal) wins a free kick on the right wing.
  Foul by Rouwen Hennings (Burnley).
  Substitution, Burnley. Michael Kightly replaces George Boyd.
  Substitution, Arsenal. Theo Walcott replaces Alexis Sánchez.
  Attempt missed. Scott Arfield (Burnley) right footed shot from the centre of the box is too high following a corner.
  Corner,  Burnley. Conceded by Mikel Arteta.
  Substitution, Burnley. Rouwen Hennings replaces Sam Vokes.
  Olivier Giroud (Arsenal) wins a free kick in the defensive half.
  Foul by Stephen Ward (Burnley).
  Laurent Koscielny (Arsenal) is shown the yellow card for hand ball.
  Hand ball by Laurent Koscielny (Arsenal).
  Attempt saved. Alexis Sánchez (Arsenal) right footed shot from outside the box is saved in the top left corner.
  Substitution, Arsenal. Mikel Arteta replaces Francis Coquelin.
  Substitution, Arsenal. Tomas Rosicky replaces Alex Iwobi.
  Alexis Sánchez (Arsenal) wins a free kick in the attacking half.
  Foul by Michael Keane (Burnley).
  Francis Coquelin (Arsenal) wins a free kick in the defensive half.
  Foul by George Boyd (Burnley).
  Substitution, Burnley. Joey Barton replaces David Jones.
target: Arsenal maintained hopes of winning the FA Cup for a third season in a row as Alexis Sanchez's goal carried them past Burnley and into round five.