The Clarets looked like taking just their second point on the road this season when Andre Gray powered home a stoppage-time spot-kick to cancel out Shkodran Mustafi's first Arsenal goal.
But, in the last of the seven additional minutes, referee Jon Moss - who had already dismissed Granit Xhaka and sent Arsene Wenger to the stands - pointed to the spot once more and Sanchez made no mistake in sealing a remarkable 2-1 win.
Mustafi had headed the hosts ahead with his first goal for the club but Xhaka's ninth red card since 2014 left the Gunners a man down as Burnley looked to fight back.
Their efforts were set to go unrewarded until substitute Francis Coquelin tripped Ashley Barnes in the box and Gray lashed home Burnley's fourth away league goal of the campaign.
Wenger was sent off for remonstrating with Moss and was not on the touchline as the referee pointed for a second penalty after Ben Mee caught Laurent Koscielny with a stray boot.
And it was Sanchez who stepped up to coolly chip home for the goal which lifts Arsenal into second spot following Tottenham's draw and Liverpool's defeat on Saturday.
Mesut Ozil drew a low save out of Tom Heaton with the hosts' first decent chance before Alex Iwobi saw his own effort deflected behind for a corner.
Koscielny headed the resulting set-piece straight at Heaton as Arsenal ramped up their efforts to establish an early lead in the contest.
Much like their Boxing Day win here over West Brom, the Gunners were being kept at bay by a staunch defensive unit, and Ozil's tame strike five minutes before the break saw Heaton make a rare first-half save.
Alexis Sanchez came within a whisker of giving Arsenal the lead before the break but could not wrap his curling shot inside Heaton's post.
Aaron Ramsey missed with a cute flick at the start of the second half before Sanchez fizzed a strike even closer to the frame of the goal than his previous effort.
The hosts wanted a penalty after a coming together between Mustafi and Gray but Moss was unmoved, but the German defender would head home his maiden Arsenal goal shortly afterwards to open the scoring.
Burnley were given some hope when Xhaka was dismissed just seven minutes later as he dived in on Defour having gifted him possession.
Moss spoke to assistant Andy Halliday and the Swiss midfielder was shown his ninth red card since 2014, which will result in a four-game ban for the summer signing.
Dean Marney was booked for a cynical foul on Ozil and was carried off on a stretcher as he injured himself in doing so, leading Dyche to introduce Joey Barton, James Tarkowski and Sam Vokes in quick succession.
And Burnley appeared on course for just a second away point of the season when Gray converted a late penalty, only for more drama to follow.
With Wenger already back in the changing rooms alongside Xhaka, Moss deemed Mee's kick on Koscielny as he attempted to clear a deep cross as a foul and Sanchez was left to produce the heroics and keep Arsenal's title aspirations alive.
Source : PA