The Reds started slowly and on the back foot, Alberto Moreno conceding a penalty after hacking down Alex Iwobi. Theo Walcott missed the resulting spot-kick but found the back of the net just 69 seconds later.
Philippe Coutinho's spectacular 30-yard free-kick just before half-time outwitted Petr Cech and changed the outlook of the game to make it 1-1. That led to a sublime start to the second-half.
Just over quarter-of-an-hour after the restart, it was 4-1 after Adam Lallana, Coutinho and debutant Sadio Mane all scored excellent goals to give the visitors a convincing advantage.
Their failure to manage the game meant that Arsenal immediately reduced the deficit, Alex Oxlade-Chamberlain's strike deflecting beyond Simon Mignolet to make it 4-2 straight from kick-off.
It was a nervy finale when Calum Chambers headed in from Santi Cazorla's free-kick to make it 4-3 - but Klopp's side did well to see out the game for an enthralling three points to start the season.
Reds' ratings from Emirates classic
Simon Mignolet: Conceded three but could do little about any of them in truth. Did well to deny Walcott from the penalty spot, guessing the right way and parrying the ball clear of danger, and yet had very few other saves to make. Well beaten for the first goal, defeated by a deflection for the second and let down by his defence for the free-kick. 7/10.
Nathaniel Clyne: His cross for Liverpool's third goal was excellent, driving the ball into the perfect area for Coutinho to beat Cech and make it 3-1, and otherwise he produced a relatively trademark performance. Solid if unspectacular, he was regularly on hand to support the attack in the first-half, and was sound defensively. Just the same in the second 45 minutes. 7/10.
Ragnar Klavan: One second-half moment, in the 70th minute, summed up his game. Caught on the back foot against Alexis Sanchez, he did superbly to muscle the Chilean out and win the ball back for his side. Solid and dependable, the 31-year-old's measly fee looks a snip on this evidence. Carried his impressive pre-season form into the real thing with ease, although could have done better in preventing Chambers score. Booked. 7/10.
Dejan Lovren: Less impressive than his centre-back partner, albeit not exactly the erratic performance Lovren produced earlier in his Anfield career. Was unfortunate with Arsenal's second, which took a nick off him to jump up over Mignolet's outstretched right arm, but looked unconvincing when forwards ran at him. Was shown inside out by Alexis Sanchez at one point and was booked for bringing him down in revenge seconds later. Good on the ball, though, just not as solid as he looked at times towards the end of last term. 6/10.
Alberto Moreno: Easy to see why supporters are so frustrated with the Spaniard off of performances like this. Almost gifted Aaron Ramsey a great chance, heading a long ball into the midfielder's path before making amends, albeit with a forceful sliding tackle in his own box. Gave away a needless penalty by hacking down Walcott and was caught too far forward for Walcott's goal just a couple of minutes later. Wasn't solely his fault, with Lallana guilty of cheaply losing possession and no-one covering, but didn't recover well enough. Much better when Liverpool were on the front foot second-half, but time will tell whether he has a long-term future at the club. Booked. 4/10.
Jordan Henderson: The deepest midfielder of a three, the Englishman often dropped into a defensive midfield position out-of-possession when Liverpool took on a 4-1-4-1 shape. Wasn't his usual robust-self as a result, but did his job and improved when Can was brought on and he was given licence to move further forward. Good on and off the ball. 7/10.
Adam Lallana: Good example of Liverpool's 'Jekyll and Hyde' performance with a below-par display in the first 45 before a much-improved showing after half-time. Looked sluggish and off-pace and was too easily dispossessed for the first goal, when Moreno had come forward anticipating a counter-attack. But took his goal expertly in the second-half, controlling Wijnaldum's chipped ball and dispatching well. Coutinho's goal was the game-changer but he inspired a rampant spell which saw Liverpool run away with it. 6/10.
Georginio Wijnaldum: Looked out-of-place in a central midfield role in the first-half, though that could have been due to Liverpool's system as a whole. Wasn't imposing or physical enough as Arsenal regularly bypassed the Reds' midfield, but was better in the second-half and registered a fine assist to set up Lallana, eyeing him across the other side of the box with a fine cross. Should have done much better with a first-half opportunity, where he sidefooted an effort straight at Cech after a promising counter. 6/10.
Sadio Mane: As debuts go, Mane's was as close to perfect as they come. He was Liverpool's best player in the first-half, when most of his teammates weren't in the game. His pace, purpose and driving runs at the defence were the only source of danger from Klopp's men and he continued as such in the second-half. His goal was exactly why Klopp spent so much money to bring him in, powering down the right before cutting into the box and curling a left-footed strike into the far top corner in exquisite style. Just about edges out Coutinho for the best player on the pitch for how consistent he was throughout. Can be a big, big player for Klopp's men this year if he repeats this kind of performance on a regular basis. 9/10.
Philippe Coutinho: There's good reason why fans and teammates alike refer to him as the 'Little Magician', you know. Having had a quiet first-half, his sensational long-range free-kick on the cusp of half-time completely changed the game. It was his first ever free-kick goal in the Premier League, but you wouldn't have thought it, such was the way it curled into the top corner from 30-yards. He exploded into life after the interval and added a second goal for good measure, connecting with Clyne's cross to beat Cech from point-blank range. His afternoon was cruelly cut short with 20 minutes remaining and Klopp and co. will hope that was only because of cramp. One of Liverpool's stand-out players, as he so often can be, and earned his Man of the Match award for how he inspired the rest of the team to such a thrilling win. 8/10.
Roberto Firmino: Slow and abject first-half, there wasn't much evidence of the fantastic kind of link-up with Coutinho he's showed previously. But like others, he was much more lively after the break and got involved in all of the Reds' combination play. Wasn't quite as influential as he can be, but his work ethic was typically industrious. 6/10.
Emre Can (Coutinho 69'): Surprisingly left out, Can slotted in effortlessly after replacing Coutinho and did his duty commendably. Ensured the Reds' midfield weren't overrun and added more balance, control and composure. 6/10.
Divock Origi (Lallana 75'): Hold-up play was impressive. Helped give his side someone to hang the ball up to when the back-line needed a breather. 6/10.
Kevin Stewart (Wijnaldum 87'): Brought on to run the clock down and add some more legs and composure in the middle. N/A.
Jürgen Klopp: Whatever he said at half-time was obviously exactly what his team needed. A much-needed injection of impetus, the performance produced between the start of the second-half and the hour mark was what was expected from Liverpool as opposed to the dour first-half display that was masked by Coutinho's free-kick. Their hunger, fluidity and drive was too much for Arsenal as they ruthlessly tore through their back-line - but question marks remain defensively. Arsenal scored three without posing too many problems, particularly in the second-half. Plenty of positives, but also plenty to work on. Improvement likely to come with returning players and maybe even a new left-back. 7/10.