The hosts took the lead through Joel Matip towards the end of a first half in which Arsenal had chances despite ceding the lion's share of possession and territory and then took the game away from their visitors during a dominant second half.
Mohamed Salah doubled the advantage from the penalty spot four minutes after the interval and then sealed the points in style with a fine individual effort to ensure that Jurgen Klopp's side are the only team with a perfect record after three league games this season.
A first clean sheet of the campaign did elude the European champions as Lucas Torreira came off the bench to score a consolation, but Liverpool were worthy of a club-record-equalling 12th successive league victory to move out on their own at the top of the table.
Liverpool made a flying start to the contest as they looked to make the most of a narrow Arsenal diamond in midfield, affording Andrew Robertson the space to drive forward and flash a low cross into the box which was just too far in front of Roberto Firmino.
Georginio Wijnaldum then snatched at a half-chance inside the area as Arsenal struggled to get out of their own half inside the opening 10 minutes, although Unai Emery's gameplan did soon become clear as they looked to release the pacy front two of Pierre-Emerick Aubameyang and full debutant Nicolas Pepe.
One such break almost saw Arsenal take the lead against the run of play when Adrian raced off his line but could only clear the ball as far as Aubameyang, who lifted his finish just wide of the empty net with the Liverpool keeper in no-man's land.
The chance did seem to give the visitors a much-needed boost in confidence and they soon began to grow into the game with Pepe drawing the first save with a strike from range after 19 minutes.
Liverpool carried an ever-present threat themselves, though, and could have taken the lead midway through the first half when Dani Ceballos's risky pass into his own area went straight to Sadio Mane, who took a touch but was then thwarted by Bernd Leno.
Pepe was looking the most dangerous for Arsenal and came close to breaking the deadlock shortly after the half-hour mark when he collected the ball at the end of a sweeping move and looked to pick out the top corner, putting his curling 25-yard strike narrowly wide.
An even better chance arrived for the club-record signing three minutes later as he streaked clear on a counter-attack, skipping past Robertson to leave him with only Adrian to beat but then putting his finish straight at the Liverpool keeper when he should have scored.
It proved to be a costly miss for the Gunners as Liverpool took the lead shortly afterwards, but not before Salah had come close with a shot on the turn which fizzed just wide of the far post.
When the opener did arrive it came from the more unlikely figure of Matip, who attacked Trent Alexander-Arnold's corner really well and powered his header into the top corner while the Arsenal defenders were busy focusing on the threat of Virgil van Dijk.
Liverpool perhaps should have been two ahead going into the break when they created another chance from a corner, but this time Mane got his header all wrong and failed to test Leno despite being unmarked.
Nonetheless, Arsenal will have gone into half time feeling that there was still something in the game for them - a hope which was effectively extinguished within four minutes of the restart.
Deadline-day signing David Luiz was penalised for an obvious tug of the shirt on Salah inside the area, and the Liverpool talisman proceeded to pick out the top corner with a perfect spot kick which left Arsenal with a mountain to climb.
The visitors did squander a golden chance to pull a goal back and restore hope within three minutes when Aubameyang raced through on goal, only for the striker to take too long over his finish and allow Matip the chance to put in a fine last-ditch challenge.
It was a rare moment of attacking threat for Arsenal in the second half, though, and Liverpool were immediately back on the front foot with Alexander-Arnold only being denied by a good block from Sokratis Papastathopoulos.
Salah then nodded over from a Robertson cross when he should have perhaps done better, but the two-time Golden Boot winner more than made up for that moments later with a stunning solo effort to well and truly put the game to bed.
The winger skipped past Luiz all too easily before darting into the box and stroking his finish into the bottom corner for his third league goal of the season in as many games.
At that stage it looked to be a question of how many Liverpool would go on to get as a gulf in class became evident, and Leno was needed to deny Firmino at the near post shortly after the hour mark.
Focus began to switch to a clean sheet as the second half wore on, but last season's stingiest defence were left still waiting for a first shutout of the new campaign when Torreira pulled one back five minutes from time.
Aubameyang fed the ball to the diminutive midfielder, who saw his initial shot smothered by Henderson but then converted after it fell kindly for him to have a second attempt.
Liverpool comfortably saw the win out without any further scares to move three points clear at the top of the embryonic Premier League table, making it 12 top-flight wins in a row to equal the club record set by Kenny Dalglish's side from April to October 1990.
LIVERPOOL (4-3-3): Adrian; Alexander-Arnold, Matip, Van Dijk, Robertson; Henderson, Fabinho, Wijnaldum (Milner 69'); Salah, Firmino (Lallana 86'), Mane (Oxlade-Chamberlain 77')
ARSENAL (4-3-1-2): Leno; Maitland-Niles, Sokratis, Luiz, Monreal; Guendouzi (Mkhitaryan 86'), Xhaka, Willock (Lacazette 81'); Ceballos (Torreira 61'); Pepe, Aubameyang