The Gunners incredibly battled back to turn the game on its head, finding the net through Alexis Sanchez, Granit Xhaka and Mesut Ozil, but the sixth and final goal would go Liverpool's way as Roberto Firmino struck to earn his side a point.
At the end of a topsy-turvy affair at the Emirates Stadium, Liverpool remain one point and one place above their opponents in the top-four race.
The visitors were forced into a change inside the opening 13 minutes as skipper Jordan Henderson, used from the off in 15 of their 19 league games, hobbled off with a hamstring problem to be replaced by James Milner.
Having dominated much of the possession up until that point, Liverpool slowly started turning it into chances as they took complete control of the match.
Coutinho provided a warning of sorts 16 minutes in with a wayward shot, before two quick-fire chances for Firmino at the end of left-sided crosses.
Liverpool did not have to wait too much longer for a breakthrough, however, as Cech failed to keep out Coutinho's deft header after Salah's cross deflected nicely into his path.
That made it 54 goals scored in all competitions by Coutinho, Salah, Firmino and Sadio Mane this term, and the latter nearly added to that tally when picking out Cech with a deflected shot.
The Reds looked good value for a second as the first half edged to its conclusion, with Firmino's effort fizzing across the roof of the net before two big openings for Salah and Mane went begging.
Laurent Koscielny, who perhaps should have done better to close down Salah in the build-up to the opener, slipped at a costly moment and Salah was clean through.
Cech kept out the shot but Mane had a big target to aim for from the follow-up, only to attempt an audacious overhead kick and miss completely.
Seven minutes into the second half Liverpool found what looked at the time like being an all-important second goal of the match, courtesy of a 15th of the season for Salah in the top flight after curling a shot past Cech via a touch from substitute Shkodran Mustafi.
In the space of 388 seconds, though, Arsenal went from two down to 3-2 up in a remarkable comeback, beginning with a powerful Sanchez header at the back post when getting ahead of Joe Gomez.
The second came from a well-struck Xhaka drive, which served through the air but should have been dealt with by Mignolet, who saw it fly right past him down the middle.
One of the most deadly fightbacks in Premier League history was completed seconds later when Ozil and Alexandre Lacazette exchanged passes for the former to dink the ball over Mignolet from close range.
Mane volleyed over the back post and Lacazette turned the ball over the bar from a good position at the other end, as both teams continued to attack.
Liverpool would get the sixth goal, however, with Cech this time having a howler by failing to get enough on Firmino's shot, helping it into the air but not over the crossbar.
Former Arsenal man Alex Oxlade-Chamberlain was introduced for the final six minutes, welcomed on to the field with a mixture of polite applause and jeers from the home supporters.
This fixture has seen more goals in the 90th minute or later than any other in Premier League history - with 17 of them in all - and Salah came closest to winning it late on with his attempt ending in the side-netting.
Arsenal: Cech; Bellerin, Koscielny, Monreal (Mustafi 46'), Maitland-Niles; Xhaka, Wilshere; Iwobi (Welbeck 78'), Ozil, Alexis (Walcott 89'); Lacazette
Liverpool: Mignolet; Gomez, Lovren, Klavan, Robertson; Henderson (Milner 13'), Can, Coutinho (Oxlade-Chamberlain 84'); Salah, Firmino, Mane (Wijnaldum 80')