Youri Tielemans gave Leicester the lead after just six minutes, but the visitors recovered well and took control of the game.
David Luiz brought the Gunners level when he headed home Willian's free-kick before Alexandre Lacazette put them ahead. He converted from the penalty spot after VAR had ruled that Wilfred Ndidi had blocked Nicolas Pepe's shot with his arm.
Pepe himself then made sure of the win when he tapped home from close range after Willian's through ball deflected into his path.
The victory sees Arsenal climb the table to ninth and they now have West Ham United in their sights in fourth place, while Leicester are in third.
Once again, Mikel Arteta's men showed great resolve. Just like in midweek against Benfica in the Europa League, Arsenal displayed good character and recovered well after going behind. The Gunners have often been criticised for having a soft underbelly, but they not only survived this early scare, they overcame it to dominate.
The injury to Emile Smith Rowe will be a concern. Arteta rested several players in this game but not Smith Rowe, who has looked leggy recently. It's unclear how long he'll be out for. Also, Arsenal's start was abysmal, with Tielemans having all the time and space in the world to open the scoring.
Manager rating out of 10
MIKEL ARTETA 8 - Arteta rotated his players and rested Bukayo Saka, Arsenal's key player in recent weeks, and Pierre-Emerick Aubameyang. The risk looked to have backfired in the early stages, but his side came good and he'll be delighted with how his selection responded. Not only that, but their attacking play was sublime at times.
Player ratings (1-10; 10 = best. Players introduced after 70 minutes get no rating)
GK Bernd Leno, 6 - He'll certainly have had far more comfortable starts to an afternoon. The experienced keeper has been Arsenal's saving grace at times, but he almost presented Jamie Vardy with a tap-in just seconds after his side had fallen behind, while he'll have been relieved that he parried Harvey Barnes' swirling strike forwards and not into the net.
DF Cedric Soares, 7 - Produced a well-timed tackle to stop Marc Albrighton in his tracks as Leicester looked to make a fist of it at 3-1 down. He looked to unleash Pepe a couple of times too with Arsenal well and truly in the ascendency.
DF David Luiz, 7 - Hauled Arsenal back into the game with a bullet header from Willian's free-kick and he provided another important contribution when he thwarted Jamie Vardy at the expense of a corner, which he headed clear.
DF Pablo Mari, 6 - The defender was all over the place as Leicester started the game with a bang. The response was excellent though and he did just about enough to nudge Vardy over as the forward looked to race through on goal. That could have been a calamitous moment, but referee Paul Tierney deemed the challenge to be a fair one.
DF Kieran Tierney, 6 - He really has grown into a vital cog within this Arsenal side and he was once again at the heart of some of his side's best attacking play with some electric bursts forward down the left.
MF Mohamed Elneny, 6 - Almost found himself caught up in a moment of madness when his attempted clearance cannoned off Kelechi Iheanacho and spun into the side netting with Leno scrambling. Aside from that he was steady until his withdrawal in the 68th minute.
MF Granit Xhaka, 6 - Found himself chasing the heels of Tielemans in vain as the Dutchman got the Foxes off to a flyer. From then on he could only get better though, and he picked out Willian who dragged wide before playing his part in a remarkable third Arsenal goal.
MF Nicolas Pepe, 8 - The Ivory Coast international has been criticised at times, but this was a positive display from the winger and his quick feet frequently tormented Luke Thomas. He won the free-kick that saw his side on level terms and finished off an exquisite free-flowing move as the Gunners put the game to bed.
MF Emile Smith Rowe, 7 - Arsenal could have been 2-0 down within seven minutes, but that did little to deter the youngster who always opened the hosts up with a brilliant charging run of his own but Lacazette couldn't get on the end of it. Big concerns for Arteta though as Smith Rowe limped off just before the break.
MF Willian, 8 - Quite possibly his best display in an Arsenal shirt. At 32-years-old the winger still covered every bit of turf in a tireless outing. His pinpoint delivery was completely on the money as Luiz powered the Gunners level and he played an important role in both goals that followed.
FW Alexandre Lacazette, 7 - The forward didn't actually see too much in the way of clear cut chances. The best strikers only need one opportunity though and, while it always looks easier from 12-yards, he showed ice-cool composure to send Schmeichel the wrong way from the penalty spot.
MF Martin Odegaard, 8 - Forced on just before half-time and made an immediate impact, feeding Pepe on the right whose effort resulted in a spot kick which Lacazette duly converted. He had a big hand in the third too, keeping the ball alive with remarkable feet to somehow find Willian.
MF Thomas Partey, 6 - Replaced Elneny in the 68th minute. Tried his luck with an ambitious effort from 35 yards but the ball struck Ndidi and looped behind for a corner.
FW Pierre-Emerick Aubameyang, NR - Replaced Lacazette in the 83rd minute but his involvement was little more than a cameo appearance, although he was inches away from adding a fourth when he fired just wide from outside the box.