The Gunners won both matches last season by virtue of last-gasp goals - a Sanchez spot-kick also proving decisive at Emirates Stadium in January - and the Chile international repeated the feat after James Tarkowski was adjudged to have pushed Aaron Ramsey.
Having won just one of their previous six Premier League away trips, Arsenal appeared set to drop more points on the road after an uninspiring performance in attack.
Arsene Wenger's side clearly missed the influence of Mesut Ozil, sidelined by illness, as they struggled to unlock a Burnley outfit which had previously conceded only twice in six home league matches this term.
But the last-gasp spot-kick proved all the visitors needed to leapfrog rivals Tottenham into fourth, with Sean Dyche's side missing their own opportunity to do exactly that.
Instead, Dyche's men were left to contend with yet another heartbreaking defeat to a team against which they have now lost 1-0 in each of the past three Turf Moor meetings.
Burnley began with typical vigour as Ashley Barnes, starting ahead of Chris Wood, looked an early danger in the air.
Their energetic start came close to earning the opener when the ball was worked right to Johann Berg Gudmundsson, who danced around Nacho Monreal and slammed his shot against the outside of the near post.
Ramsey should have broken the deadlock after Alexandre Lacazette escaped down the left and whipped in a teasing cross, but the midfielder skied his half-volley under little pressure.
An entertaining first half-hour continued with Petr Cech preventing Jeff Hendrick's header from going back across goal, while Monreal drove a long-range effort marginally wide at the other end.
Cech remained the busier of the two goalkeepers late in the half as he pushed away Robbie Brady's goal-bound free-kick to keep it scoreless at the interval.
PIC: James Tarkoswki makes an important block. pic.twitter.com/XB8jd9aO1F— Burnley FC (@BurnleyOfficial) November 26, 2017
Arsenal appeared to have more purpose immediately after the restart and saw Ben Mee block a good opening for Sead Kolasinac, before Lacazette forced Nick Pope into his first real save.
Gudmundsson did fire another warning shot at Cech but the visitors had wrestled the initiative by the hour and saw appeals for a penalty waved away after Brady slid in on Hector Bellerin.
The Republic of Ireland international then threatened to snatch the winner by firing over the bar with just over 15 minutes to play.
Burnley's nerves were tested when Wilshere's attempt at the other end deflected wide off Tarkowski, who cost his side a share of the points in the second minute of added time as Ramsey went down and Sanchez squeezed the ball beyond Pope's fingertips into the bottom-left corner.
Key Opta stats:
- Arsenal extended their unbeaten run to 10 top-flight matches against Burnley (W7 D3).
- Arsenal will end the day inside the top four of the Premier League outside the month of August for the first time since March this year.
- Alexis Sanchez has scored four goals in his last five Premier League appearances against Burnley – he's only scored more against Sunderland (6) and Hull (7).