The Foxes took the lead after just 14 minutes when Kelechi Iheanacho fired home, and the Nigerian striker was heavily involved again just one minute later when Konstantinos Mavropanos dragged him to ground and received his marching orders as a result.
The Gunners managed to avoid further damage before half time despite their numerical disadvantage, and even levelled the scores in the second half when Pierre-Emerick Aubameyang continued his recent good form in front of goal.
However, a Jamie Vardy penalty 14 minutes from time restored Leicester's lead before Riyad Mahrez capped off the scoring in the final minute of a pulsating match as Arsene Wenger fell to his first ever defeat at the hands of Leicester in his penultimate match in charge of Arsenal.
The North Londoners incredibly remain the only team in England's top four divisions and Europe's top five leagues yet to pick up a point away from home in 2018, losing all seven so far this calendar year to record their worst run of away form since 1966.
Leicester, meanwhile, end a five-match winless streak to ease the pressure on manager Claude Puel and guarantee a top-half finish this season.
It was actually Arsenal who started the brighter with Alex Iwobi forcing Eldin Jakupovic into an early save at the end of a slick team move, but that was about as good as it got for the visitors in the first half.
Leicester had made another sluggish start, but any fears that they would suffer from a similar malaise which was so poorly received by their fans during Saturday's defeat to West Ham United were soon put to rest.
Arsenal were unable to clear their lines despite having a number of chances to do so, and eventually a cross to the back post found Fousseni Diabate who turned it back into the middle for Iheanacho to drill past Cech.
Things went from bad to worse for the Gunners just a minute later when the lively Iheanacho nicked the ball off Mavropanos, who subsequently dragged the Leicester striker to ground in a challenge deemed worthy of a straight red card due to his being the last man.
Suddenly in the space of one minute Arsenal found themselves a goal and a man down, and it looked as though it would get even worse before the half was out as Leicester began to swarm all over their visitors.
The free kick which followed Mavropanos's sending-off saw Mahrez float the ball in to an unmarked Harry Maguire, who met it with a thunderous volley which was turned over the crossbar by a fine reaction stop from Cech.
Arsene Wenger quickly turned to his bench in an attempt to limit the damage of the red card as Danny Welbeck was sacrificed for Shkodran Mustafi, but that did little to stem the relentless tide of Leicester attacks.
Arsenal's defence backed off Iheanacho as he collected the ball in space before advancing towards the edge of the area, but this time Cech was equal to the striker's effort.
Another defensive error provided Silva with a shooting opportunity which he wasted on the half-hour mark, and Leicester were soon given a warning shot of why a second goal was so important when Arsenal finally began to fight back.
Henrikh Mkhitaryan drew a smart low save from Jakupovic with his first-time effort from an Iwobi pass before the resulting corner eventually fell at the feet of Rob Holding, who poked the ball agonisingly wide of the target with the Leicester keeper stranded.
Leicester were soon back on the front foot again, though, with Mahrez firing one effort off target before a clever free kick from Silva went under the wall and forced another low save from Cech, who saw it late.
Cech was by far the busier of the two goalkeepers in the first half, and he came to Arsenal's rescue again moments later as the non-stop action continued, saving from a low strike from Iheanacho with just four minutes remaining in the first half.
Iheanacho continued to look the most likely for the hosts, and there was still time for him to have two more sights of goal before the interval as first he saw the ball nicked off his toes just as he was about to pull the trigger from close range, and then he tried his luck from inside the centre circle after once again catching an Arsenal player in possession.
There was finally time for both sets of players to catch their breath when the referee blew for half time at the end of an action-packed opening 45 minutes, and Arsenal will have been pleased to go into the interval with only a one-goal deficit.
Indeed, the Gunners almost wiped out that deficit within two minutes of the restart as the second half quickly picked up from where the first had left off, with Sead Kolasinac ghosting into the box before thumping an effort against the foot of the post from a tight angle.
Leicester were once again up the other end moments later, though, and Cech got just enough on Diabate's effort when one on one to take the sting off it before Mustafi completed the clearance off his own goalline.
It looked like shaping up to be another busy half for Cech when he kept out a Mahrez free kick in the 52nd minute, but the end-to-end nature of the match was always likely to expose the frailty of a one-goal lead and that proved to be the case just a minute later as Arsenal levelled things up.
Ashley Maitland-Niles burst down the right flank and reached the byline before cutting the ball back for Aubameyang, who was initially denied by a stunning save from Jakupovic before reacting quickest to the rebound to fire it into the roof of the net - his ninth goal in just 12 Premier League appearances for the club.
In the true spirit of the breathless thriller the match had become, Leicester were immediately back on the front foot, and Iheanacho fired another effort off target before Christian Fuchs scuffed one of his own wide from the edge of the area.
The match eventually hit an inevitable lull around the hour mark, but that did not last long and Mkhitaryan almost blasted his side in front with just over 20 minutes remaining when his stylish touch was followed by a powerful half-volley which flew over the near post.
Arsenal's refusal to sit back with 10 men was every bit as integral to the flow of the game as Leicester's own desire to make the most of their numerical advantage, but the hosts provided one counter-thrust too many for the Gunners with 15 minutes remaining when Mkhitaryan was adjudged to have brought Demarai Gray down inside the area.
Graham Scott's decision to point to the spot looked like a contentious one, but there was never any doubt about the penalty itself as Vardy slammed his finish into the top corner for his sixth goal in his last six Premier League games against Arsenal.
Aaron Ramsey had a chance to restore parity again 10 minutes from time with a header which was gathered by Jakupovic, but that proved to be their final sight of goal and Leicester finally wrapped up a thrilling victory in the final minute of normal time.
Mahrez raced on to a through-ball down the right flank before cutting inside and fooling both Cech and Holding with a dummy to leave himself with a simple finish.
It was a fitting moment of quality to end a frenetic contest, and the Leicester fans added their own touch of class in the closing stages when they joined in an applause for Wenger, who now has just one match remaining of his 22-year Arsenal reign.
That final-day fixture at Huddersfield Town will give Wenger one last chance to end Arsenal's dismal run of away form, with the Gunners failing to pick up a single point on their travels since New Year's Eve.
Leicester, meanwhile, take on Arsenal's North London rivals Tottenham Hotspur on the final day knowing that a victory - coupled with defeat for Everton - would see them end the season in eighth place.
LEICESTER (4-4-2): Jakupovic; Simpson (Dragovic 82'), Morgan, Maguire, Fuchs; Mahrez, Choudhury, Silva, Diabate (Gray 72'); Vardy, Iheanacho (Barnes 88')
ARSENAL (4-3-3): Cech; Maitland-Niles, Holding, Mavropanos, Kolasinac; Iwobi (Nketiah 84'), Xhaka, Ramsey; Mkhitaryan, Aubameyang, Welbeck (Mustafi 19')