The two sides got their 2018 campaigns off to a flying start with a pulsating contest from start to finish, although it took until the 63rd minute for either to break the deadlock when Jack Wilshere lashed home his first Premier League goal since May 2015.
That lead lasted just four minutes before Eden Hazard levelled things up from the spot, though, and it looked as though Chelsea had done enough to come away with all three points when Marcos Alonso scored with just six minutes remaining.
However, the topsy-turvy and end-to-end London derby had one final twist in the tale as Bellerin fired home a fine half-volley to earn a point for the home side, who held on despite Chelsea hitting the crossbar just seconds later.
Both teams welcomed key players back into their starting lineup as Mesut Ozil returned from a knee injury for Arsenal and Hazard was restored to the Chelsea side after being rested last time out.
The Gunners were missing Laurent Koscielny and Sead Kolasinac in addition to the likes of Nacho Monreal, Aaron Ramsey and Olivier Giroud, though, leaving Arsene Wenger's side with an inexperienced defence.
The imbalance between Arsenal's defence and attack was a feature of the first half too, with Alexis Sanchez the first to come close when his deflected free kick looped narrowly wide of the target while Thibaut Courtois was stranded.
Just seven minutes later Chelsea should have broken the deadlock when a mix-up between Calum Chambers and Petr Cech allowed Alvaro Morata to race clean through on goal, but he fluffed his lines with his finish and sent a dreadful effort well wide of the target.
Morata was close to getting on the end of a low Victor Moses cross moments later, but Arsenal were soon on the attack again in an end-to-end match and will have been wondering quite how they failed to open the scoring in the 17th minute.
Sanchez played a slick one-two with Ozil inside the box before poking his finish towards goal, only for Courtois to pull off a stunning reaction save. Chelsea still needed both posts to come to their rescue, though, with the ball hitting the inside of one upright, rolling along the line and striking the other before Courtois gratefully dropped on it.
The Chelsea keeper was called into action again shortly afterwards when he got down smartly to deny Alexandre Lacazette after the striker had turned sharply inside the box.
The game continued to ebb and flow and it wasn't long before the visitors had another chance of their own when Tiemoue Bakayoko was kept out by a good reaction stop from Cech.
Alonso was the next to threaten with a free kick which flew over the bar, but Arsenal were back up the other end just a minute later and it took another fine save from Courtois to tip Ozil's 25-yard drive wide of the post.
The final chance of a pulsating first half fell the way of Chelsea following a beguiling layoff from Hazard, but Cesc Fabregas failed to add the finish which the buildup play deserved when he curled his effort off target from the edge of the box.
Somehow the opening 45 minutes ended goalless, but the second half quickly picked up in a similarly entertaining vain with Cech making an early double-save to first deny Hazard one on one with his feet before tipping Alonso's goalbound header around the post.
Bakayoko fired a decent chance over the bar from the resulting corner before Arsenal were soon on the front foot again just two minutes later, with Lacazette being thwarted at point-blank range by the brilliant Courtois once more after the ball had fallen kindly to him inside the area.
The hosts came forward again shortly after the hour mark when Ozil played a one-two with Lacazette on his way down the left channel, only to see his square ball towards Sanchez cut out when he could have gone for goal himself.
It had been a goalkeeping masterclass at both ends of the field for the opening hour, but Courtois was finally beaten in the 63rd minute when Rob Holding's pass was deflected into the path of Wilshere, who blasted an unstoppable strike past the Belgian at his near post.
Arsenal refused to rest on their newfound lead, though, and were pushing for a second within two minutes as Courtois was called into action to deny Lacazette once again.
For the second match in a row, though, Arsenal quickly gave up their 1-0 lead courtesy of a controversial penalty as Hazard went down theatrically following a wild swing of the boot from Bellerin.
Wenger was again left incensed on the sidelines, and his mood was not improved when Hazard dusted himself off to fire the spot kick down the middle himself, restoring parity only four minutes after falling behind.
While a number of players were performing at the peak of their ability, the same could certainly not be said of Morata, who squandered another clear chance with 20 minutes remaining when he latched on to Fabregas's through-ball only to spoon his effort wide of the target.
N'Golo Kante was the next to come close with a deflected drive that span narrowly past the post, but Chelsea's increasing pressure did eventually pay off when they took the lead with only six minutes or normal time remaining.
Davide Zappacosta beat a tiring Ainsley Maitland-Niles down the right flank before drilling a low cross into the box which Alonso met with a darting run to the front post before finishing with all the aplomb of a seasoned striker.
It was another Spanish full-back who would write the final chapter of an incredible contest, though, as Bellerin rescued a point for his side in the second of four minutes stoppage time when he steered a fine half-volley into the far corner after the ball had dropped to him inside the box.
Even then the drama was not over, and Arsenal's jubilation almost turned to despair seconds later when Morata was once again played clean through on goal, but the striker made it a hat-trick of spurned one-on-ones when he fired a tame effort straight at Cech.
The rebound bounced out as far as Zappacosta, but his thunderous follow-up effort crashed against the crossbar as Arsenal hung on for a well-earned point to extend their unbeaten streak to nine matches across all competitions.
Despite that, Arsenal remain sixth in the table and five points adrift of the Champions League places, while Chelsea miss the chance to climb up to second and could now find themselves only three points clear of fifth should Tottenham Hotspur beat West Ham United on Thursday.
These two sides will now turn their attention to the FA Cup third round before quickly resuming acquaintances in the first leg of the EFL Cup semi-finals in a week's time.
Arsenal (3-4-3): Cech; Chambers (Walcott, 88'), Mustafi, Holding; Bellerin, Xhaka, Wilshere, Maitland-Niles; Ozil, Lacazette (Welbeck, 80'), Sanchez
Chelsea (3-5-1-1): Courtois; Azpilicueta, Christensen, Cahill; Moses (Zappacosta, 56'), Bakayoko, Kante, Fabregas (Drinkwater, 71'), Alonso; Hazard (Willian, 82'); Morata