Arsene Wenger's side doubled their advantage moments later when Calum Chambers prodded home to score his first goal for the club, while the points were sealed late on thanks to another strike from Sanchez.
Here, Sports Mole takes a closer look at the action between the two teams at the Emirates Stadium.
On target: 13
On target: 2
Was the result fair?
Despite Arsenal's lengthy wait to score their opening goal, there can be no denying that they were excellent value for the win. The hosts were dominant throughout, ending up with 32 attempts at goal, while Burnley threatened only occasionally on the counter-attack. Heaton put in an impressive performance for the visitors in goal, but it would have been a display of freakish proportions had he kept the Gunners out for the full 90 minutes.
Arsene Wenger's side had Burnley under pressure for pretty much the whole game, but it was only through a lack of finishing that they were made to sweat unnecessarily. The passing and movement was crisp as the hosts created chance after chance, while full-backs Calum Chambers and Kieran Gibbs offered plenty of width in high areas. However, the key element of their performance was persistence. Despite the usual disgruntlement from the home fans, the players remained positive and vibrant before eventually breaking their opponents down. With two consecutive wins in the Premier League now under their belts, Wenger will be eager for his team to put a run together in order to keep themselves challenging near the top of the table.
Another tough day at the office for Sean Dyche's men, who did well to stay in the match before collapsing late on. As expected, they had little impact on the game from an attacking perspective as Marvin Sordell and Danny Ings huffed and puffed in the final third to no avail. The visitors remained resolute in defence, soaking up pressure for much of the game, but it all fell apart for them once Sanchez made it 1-0. The biggest disappointment for the Clarets came in midfield, with the likes of Scott Arfield and George Boyd not doing enough to stop Arsenal's creative players. They needed to stem the flow in those areas, but instead they did not dig their heels in enough.
Sports Mole's man of the match
Alexis Sanchez: The Arsenal forward was a bright spark throughout, shooting on sight and giving Burnley all sorts of other problems with his pace, movement and physicality. His first goal was a superb header that any out-and-out striker would have been proud of, while he demonstrated great awareness for his second. Ten goals in his last 12 games suggests that he is finding English football very accommodating indeed.
Santi Cazorla had the goal at his mercy in the early stages of the second half, but somehow Michael Duff came across to make a last-ditch block. As good a piece of defending as it was, the Spaniard should have buried from eight yards out.
Craig Pawson was a bit too whistle-happy in the first half, stifling the flow of the game with some soft decisions. He also made a bad error during the opening 20 minutes when he failed to spot a deliberate handball from George Boyd inside the Burnley penalty area. The official improved in the second half as he allowed the game to be played at a smoother pace, which benefited Arsenal come full time.
Arsenal: The Gunners return to Champions League action on Tuesday when they host Anderlecht.
Burnley: The Clarets continue their Premier League campaign next Saturday at home to Hull City.