Kai Havertz headed the hosts in front from a corner with nine minutes of the first half remaining before he was fortunate to escape a red card for a rash tackle midway through the second on Monday (Tuesday AEST).
The victory takes Mikel Arteta's players five points clear of Manchester City who must now win at Bournemouth on Tuesday night to take the title race to the final day.
Should Pep Guardiola's side fail to win at the Vitality Stadium – where Bournemouth have not lost in the eight matches since their 3-2 defeat to Arsenal at the turn of the year – then the Gunners will celebrate the title with a game to spare.
Arsenal looked to have blown their bid to claim a first championship since Arsene Wenger oversaw the unbeaten 'Invincibles' to glory in 2004 following a 2-1 defeat at rivals City on April 19.
But the Gunners, runners-up in the league for the past three seasons, have bounced back by winning their next four without conceding, and are now in pole position to earn Arteta the second trophy of his near seven-year tenure.
Indeed, their victory here could kickstart the finest dozen days of the club's 144-year history, with a Champions League final to come against Paris Saint-Germain on May 30 – the Gunners have never won Europe's grandest competition.
Prior to then, Arteta's side will take on Crystal Palace at Selhurst Park, with, at worst, a two-point cushion over City.
Here, they huffed and puffed for most of the opening period without success. Havertz shot wide, and Leandro Trossard struck the foot of the post from outside of the area.
In the 34th minute, Bukayo Saka wanted a penalty when he fell following a challenge by Lucas Pires. Referee Paul Tierney was unmoved, and so too was VAR, much to Arteta's disbelief. But the Spaniard's touchline angst transformed to ecstasy moments later.
Arsenal had attempted two short corners before reverting to their tried-and-tested method. Saka's pinpoint cross into a congested area was met by Havertz and Arsenal had lift-off with their 18th goal from a corner in the league so far – accounting for more than a quarter of their 69 strikes.
It was also Saka's 50th Premier League assist. Only Dennis Begkamp (94), Thierry Henry (74), Cesc Fabregas (70) and Mesut Ozil (54) have managed more in red and white.
Eberechi Eze's strike clipped Max Weiss' crossbar and, while the Gunners' advantage stood only at one, Burnley were still in the game and City the title race.
Then, a huge let-off for Havertz and Arsenal when the German left a left boot high on the back of Lesley Ugochukwu's left calf in the 65th minute, but VAR elected not to get involved.
As the clock ticked down, David Raya's goal remained untested to leave the Gunners on the brink of glory.
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