Pierre-Emerick Aubameyang scored either side of half time to move out in front in the race for the Premier League Golden Boot, but the hosts were forced to endure a nervy finale after Ashley Barnes pulled a goal back for Burnley just after the hour mark.
Alex Iwobi finally put the game to bed with Arsenal's third in stoppage time, though, as Burnley's battling performance proved fruitless, consigning them to a place in the relegation zone on Christmas Day when 12 months ago they were seventh.
Arsenal, meanwhile, remain fifth and are only outside the Champions League places on goal difference - at least until Chelsea take on Leicester City later today.
The match got off to a quick start with chances at both ends inside the opening two minutes, with Ainsley Maitland-Niles first drawing a smart stop from Joe Hart before Burnley went straight up the other end and fired wide through Ashley Westwood, who should have done better.
It set the tone for a frantic opening 20 minutes or so which saw both sides look to get forward, although it was Arsenal who created the better chances and Granit Xhaka - initially back in his preferred midfield position as the hosts began the game with a diamond 4-4-2 - forced Hart into a routine save with his long-range drive in the eight minute.
Just seconds later Alexandre Lacazette stood a cross into the area for Mohamed Elneny, who was making his first league appearance of the season, but his header was blocked before a mini-scramble ensued inside the Burnley six-yard box.
The opening goal did arrive after only 14 minutes when Aubameyang fired home from Sead Kolasinac's cut-back, although the returning Mesut Ozil was the architect with an exquisite pass to Kolasinac in the first place.
Maitland-Niles then fired straight at Hart as Arsenal began to gain a greater degree of control over the game, although the early flow of chances soon diminished and the most notable moments of the rest of the first half were a couple of tussles between Barnes and Sokratis Papastathopoulos which left both players on yellow cards.
An already-injury-ravaged Arsenal defence was struck by another blow towards the end of the half as Nacho Monreal limped off to be replaced by Stephan Lichtsteiner, forcing Unai Emery to revert to a three-man defence.
The switch in formation did not appear to do Arsenal any harm initially, and having gone into the break ahead for the first time in the Premier League all season, the hosts then doubled their advantage within three minutes of the restart.
Once again Aubameyang was the scorer as he moved out on his own as the Premier League's leading marksman this season, thumping a finish into the roof of the net after Lacazette had found him at the end of a swift counter-attack.
Burnley responded well to Arsenal's second, though, and the visitors benefited from the game becoming scrappier and more feisty during the second half.
The Clarets' improvement was rewarded shortly after the hour mark when Barnes - who had been heavily involved throughout the match and was treading a disciplinary tightrope - pulled one back for his side.
The striker saw his initial effort from inside the area blocked, but a scramble eventually took the ball back to his feet and this time he made no mistake with a sharp finish past Bernd Leno - Burnley's first away league goal since November 3.
Suddenly what looked like a routine Arsenal win turned into a nervy affair for the home side, and they were fortunate that Jack Cork was wasteful with his finish when teed up in plenty of space on the edge of the box 20 minutes from time.
However, that proved to be the best Burnley could conjure in their search for an equaliser, and Arsenal finally put the game to bed in the first of five minutes stoppage time when Iwobi fired home a loose ball after Ozil's shot had been blocked, although there was a question of offside against the goalscorer.
Even so, it ended Burnley's hopes once and for all as they remain in the relegation zone, whereas Arsenal return to winning ways in their final home game of 2018 following back-to-back defeats at the hands of Southampton and Tottenham Hotspur.
Arsenal will travel to Brighton & Hove Albion for their Boxing Day fixture, whereas Burnley host Everton at Turf Moor.
ARSENAL (4-3-1-2): Leno; Maitland-Niles, Sokratis, Monreal (Lichtsteiner 37'), Kolasinac; Elneny (Torreira 59'), Guendouzi, Xhaka; Ozil; Aubameyang, Lacazette (Iwobi 78')
BURNLEY (5-3-2): Hart; Bardsley (Lowton 83'), Long, Tarkowski, Mee, Taylor; Hendrick, Cork, Westwood; Barnes (Vokes 77'), Wood (Vydra 77')