Alexandre Lacazette put Arsenal ahead after a horrendous mistake from Andy Robertson midway through the first half. But Liverpool equalised almost immediately, after Sadio Mane tucked in a rebound, when Bernd Leno parried a Mohamed Salah shot.
Robertson made amends by putting Liverpool in the lead just minutes later. The win was sealed late in the second half, as Diogo Jota volleyed in a strike on his Premier League debut.
Here's how each player fared for both teams.
Alisson Becker - 8/10
There were doubts over whether Alisson would be fit to start this game, but the Brazilian was passed fit, and put in a superb display. His save off Lacazette in the second half kept Liverpool ahead, and was the perfect example of how a goalkeeper shouldn't commit himself to any action in a one-on-one situation.
Trent Alexander-Arnold - 7/10
Alexander-Arnold registered his first assist of the season, with a superb cross to find Robertson at the back post. His crossing was great, his defensive discipline was better. Between him and Gomez, they neutralised Aubameyang completely.
Joe Gomez - 8/10
This was a first-class performance from Gomez in his first game back from injury. His recovery speed is such an asset, even when teams get in behind Liverpool. That makes the Reds so hard to attack against.
Virgil van Dijk - 7/10
He had a largely comfortable evening, as the Reds really didn't face too many threatening moments from Arsenal. The Dutchman, though, will know that they need to iron out some details in their high line, as Arsenal had a couple of chance on the break in the second half.
Andy Robertson - 7/10
Robertson's evening was off to a nightmare start, when he messed up an easy clearance to gift Lacazette a goal. But he made amends with his goal to put Liverpool ahead. In general too, he had a typical energetic shift down the left flank.
Fabinho - 8/10
Fabinho carried over his form from last week, as he moved into his preferred midfield position, from centre-back. His anticipation of danger, and the timing of his tackles were a treat to watch as he put on a defensive midfield masterclass.
Georginio Wijnaldum - 6/10
He never stopped running, and was important as Liverpool controlled possession in midfield. He could've had an assist at the end of the game, if not for confusion between Salah and Diogo Jota leading to the chance being spurned.
Naby Keita - 7/10
He was very involved in a lot of the good attacking play that Liverpool mustered during the game. His ball progression from midfield, and ability to pick a pass were key for the Reds in this game.
Mohamed Salah - 8/10
No goals or assists, but Salah played a big part in all three goals. He terrorised Kieran Tierney all game. Salah should have had an assist though, when picked out Jota with a peach of a pass, but the Portuguese put the ball into the side-netting.
Sadio Mane - 9/10
Relentless. He gave Liverpool their equaliser with an easy finish, but his contribution went well beyond that. Mane was full of running, eager with his pressing, and an absolute nightmare to deal with.
Roberto Firmino - 7/10
The leader of the Liverpool press was on his game, as Arsenal were not allowed a moment's peace in possession of the ball. He still would love to be a lot more decisive when in possession of the ball though.
James Milner - 6/10
Milner replaced Keita, as Klopp looked to calm things down in midfield, and help his side hold on to their lead.
Diogo Jota - 7/10
A missed sitter, a chance stolen off him, and then a goal - not a bad Premier League debut for Jota, was it? Perhaps more important for Liverpool is that he already looks like a hand-in-glove fit to their system.
Takumi Minamino - N/A
Came on in second-half injury time to replace Firmino.