History was repeated as Arsenal hit the Clarets with another late blow in the Premier League.
The battle for fourth spot was heading towards a goalless draw when the fourth official held his board aloft to signal two minutes of time added on but the visitors won it with the last kick for the third game in succession.
Sanchez, who netted from the spot in the 97th minute at the Emirates last term, was the man to deny Sean Dyche's side once again, stroking the ball in to the corner from the penalty spot after Aaron Ramsey went to ground under the challenge of James Tarkowski.
It was a cruel finish for the home side who were more than deserving of a point at Turf Moor but, just like last season's end, when Laurent Koscielny won it in controversial fashion, the Clarets left the field deflated.
Petr Cech was the busier of the two goalkeepers as the Gunners failed to register a shot on target in the first half.
The 35-year-old former Chelsea stopper showed exactly why he's considered to be one of the greatest goalkeepers in Premier League history when turning Johann Berg Gudmundsson's shot on to the post.
The Czech Republic international, who has won the Golden Glove accolade on four occasions, and voted the best keeper in Europe three times, did incredibly well to get a fingertip to the ball once the winger had side-stepped Nacho Monreal.
Burnley started brilliantly, stepping on to the front foot, asking questions of the away side's backline, and testing whether Arsenal could indeed do it on a cold and wet afternoon in East Lancashire.
Arsene Wenger's men almost provided a definitive answer when Alexandre Lacazette broke away down the left hand side, showing Tarkowski a clean pair of heels, before picking out Ramsey in the box but the unmarked Welshman missed the target by some margin when taking the chance first time.
The visitors went close again when Monreal was invited to pick his head up and try his luck from 25 yards out but the defender's attempt flashed wide of the upright having gathered pace on the wet surface.
Brady has beaten both Andy Lonergan and Thibaut Courtois with wonderfully executed free kicks at Turf Moor and once Laurent Koscielny brought down Ashley Barnes 20 yards from goal the Republic of Ireland international had the chance to add Cech to his list.
However, the Champions League winner telegraphed the attempt and palmed clear after Brady had lifted the ball up and over the wall.
Arsenal closed the half with a final opportunity which fell to Sead Kolasinac but the Bosnian full back headed Ramsey's cross wide having found space in-between Ben Mee and Stephen Ward.
The Gunners did hit the target early in the second half courtesy of £47m summer signing Lacazette but Pope got down low to gather comfortably.
Gudmundsson struck his attempt with far more conviction than the Frenchman at the opposite end when cutting inside from the right but, again, Cech was equal to the effort.
The hosts were defending admirably once again, their organisation, discipline and endeavour shining through, and it seemed that Arsenal would need a stroke of luck if they were to find a way through.
They almost got it when Lacazette's deflected shot had Pope scrambling to his post and again when substitute Jack Wilshere's strike flicked awkwardly off the boot of Tarkowski and sailed wide of the upright with Pope rooted to the spot.
Steven Defour's shot didn't trouble Cech as it rose over the bar and at that point the Clarets seemed destined to add another point to their tally.
But history repeated itself and the Gunners left with maximum points again. It happened when Koscielny won a penalty from an offside position to claim victory late on at the Emirates last season, it happened when Alex Oxlade-Chamberlain fired the ball in off the hand of the centre back, again in an offside position, at Turf Moor and now it was Ramsey's turn.
The midfielder threw himself to the turf animatedly when feeling a push in the back from Tarkowski and referee Lee Mason didn't need any encouragement to point to the spot in stoppage time.
Pope guessed the right way but Alexis Sanchez squeezed the ball in to the corner.