An early goal proved decisive in a tight contest at a windy Turf Moor.
Aaron Ramsey’s finish into the roof of the net within the first quarter of an hour handed title-chasing Arsenal their eighth consecutive Premier League win.
And despite the best efforts, battling Burnley were unable to find the breakthrough and stayed two points adrift of safety in the league table.
Sean Dyche named an unchanged side for the fourth consecutive game.
Meanwhile, Matt Taylor was in the matchday squad for the first time since August after recovering from an Achilles problem.
Arsène Wenger made no changes to the Arsenal starting line-up following a 4-1 victory over Liverpool last weekend.
Both teams threatened right from the start.
Alexis Sánchez was first to bring a comfortable save out of Tom Heaton with less than two minutes on the clock.
In response, Gunners goalkeeper David Ospina had to race off his line quickly to deny Sam Vokes after the Welshman outpaced Per Mertesacker down the left channel.
Moments later, the ball fell to Sánchez inside the Burnley box, but the Chilean couldn’t keep his shot on target.
However, Arsenal soon took the lead, despite some brave defending by the Clarets.
On a quick counter-attack, Heaton did brilliantly to deny Mesut Özil from a tight angle and two other efforts were blocked before Ramsey turned the ball home from close range on 12 minutes.
Burnley did not seem affected by the opener, though.
After Barnes won a free-kick in a promising position, Kieran Trippier forced a flying save from Ospina midway through the first half, as the Clarets began their search for an equaliser.
Shortly afterwards, Santí Cazorla struck a free-kick inches wide of target at the other end.
The match then became more of a scrappy affair, with both teams committing a total of 19 fouls before half-time.
After the restart, it was Burnley’s turn to cause danger again when Ospina needed to be alert to keep out Barnes’ dangerous effort on the turn.
In response, Sánchez brought another save from Heaton following a quick passing combination by the visitors.
But the game still couldn’t quite come back to life, with both teams lacking spark in their respective final thirds of the pitch.
The Clarets kept probing though and after Ben Mee got past Héctor Bellerín on the wing, George Boyd just lost his footing when the left-back’s low ball in found the Scot unmarked at the far post midway through the second half.
That moment seemed to finally liven things up and when Danny Ings forced another save from Ospina with a spectacular, overhead kick, the Turf faithful could sense an equaliser coming.
But despite late pressure and a stoppage time cameo of Taylor, Burnley were unable to find a breakthrough against an Arsenal side that closed the gap to league leaders Chelsea to four points at the top of the Premier League table.
Burnley: Heaton; Trippier, Duff, Shackell, Mee; Boyd, Arfield, Jones (Taylor 90+3), Barnes; Ings, Vokes.
Subs not used: Gilks, Keane, Ward, Ulvestad, Kightly, Jutkiewicz.
Arsenal: Ospina; Bellerín, Mertesacker, Kościelny, Monreal; Coquelin, Cazorla; Ramsey, Özil, Sánchez (Chambers 90+3); Giroud (Welbeck 81).
Subs not used: Szczęsny, Gibbs, Flamini, Rosický, Walcott.
Goals: Ramsey 12
Bookings: Mee (25’), Duff (26’)
Referee: Mike Dean
Attendance: 20,615 (3,421)