The lively Chile forward powered in a header on 70 minutes having earlier been denied by fine saves by busy goalkeeper Tom Heaton and a string of blocks by the dogged Clarets defence.
England defender Calum Chambers netted a second two minutes later, lashing in his first Arsenal goal following a corner, with both Heaton and the woodwork saving Burnley futher after Theo Walcott had made a welcome return off the bench from a long-term knee injury.
Sanchez, so impressive since a £30million summer move from Barcelona, waltzed in a third during stoppage time as the Gunners moved above West Ham on goal difference, and level with Manchester City, who play on Sunday.
As often in such fixtures, it was Arsenal, who made all of the early running, but found Burnley's two banks of rigid defence difficult to break down.
Danny Welbeck weaved into the penalty area, riding a couple of challenges, before his goal-bound shot cannoned behind off Kieran Trippier.
On 20 minutes, the Arsenal players were incensed when a corner from the right by Santi Cazorla was pushed away by the fist of George Boyd, who had not long been cautioned for a foul, but was missed by the officials, with the assistant referee David Bryan perfectly placed on the near side.
Arsenal continued to press and Cazorla flashed a 10-yard drive wide following Alex Oxlade-Chamberlain's cut-back.
Burnley, looking to get off the foot of the table with a first league win, remained under siege as the busy Sanchez got away down the left and his curling shot was parried out by Heaton, with the follow-up effort from Cazorla blocked.
Sanchez whipped another 20-yard drive just wide of the top-right corner before the Chile forward was denied by a fine reaction save from Heaton after darting into the left side of the Burnley penalty area.
There was a rare moment of panic in the Arsenal box at the start of the second half as Danny Ings squirmed clear, but between them Per Mertesacker and goalkeeper Wojciech Szczesny eventually managed to mop up the loose ball.
The Clarets, in a black away strip, continued their improvement and enjoyed a spell of possession, pushing Arsenal back.
But slowly the hosts regained some momentum, with Sanchez and Oxlade-Chamberlain the main agitators.
Somehow Burnley survived when Sanchez chipped the ball down the right for Chambers to run onto and cut back across the six-yard box, where Cazorla's close-range shot was blocked brilliantly by Michael Duff.
The visitors' resistance was, though, finally broken on 70 minutes.
A deep ball in from Chambers hung up at the far post, where Sanchez powered between Duff and Trippier to crash a header into the bottom-right corner.
Before Burnley could regroup, it was 2-0.
A corner from the left came through to Welbeck, whose shot was blocked on the line, with Chambers quick onto the rebound, lashing it in from two yards to celebrate a first Arsenal goal since his summer move from Southampton.
Arsenal made a double change with 10 minutes left as Lukas Podolski replaced Welbeck and Walcott came on for Oxlade-Chamberlain, almost 10 months after being carried off in an FA Cup tie against Tottenham with a knee injury which would shatter his World Cup dream.
Heaton saved brilliantly from an acrobatic effort by Podolski, and then turned Walcott's angled drive onto the crossbar before the German saw a fierce left-footed volley crash off the post.
Arsenal chalked up a third in stoppage time when the ball was played through the Burnley box to Sanchez, who calmly weaved past two defenders and stabbed the ball in from close range.
Source : PA
Source: PA