It was only Liverpool's second win at Arsenal in 21 attempts and put an early dent in Arsene Wenger's hopes of leading the London club to a first league title since 2004.
The Frenchman, who celebrates his 20th anniversary as Arsenal manager in October, has now seen his team beaten at home in their opening league game three times in four years.
Klopp, starting his first full season as Liverpool manager, celebrated each of the visitors' goals with trademark abandon, but blamed his reaction to Mane's goal for the way his side eased off at the end.
Injuries to Per Mertesacker and Gabriel, plus Laurent Koscielny's lack of match fitness, obliged Wenger to deploy 21-year-old Chambers and 20-year-old new boy Rob Holding at centre-back.
Liverpool, meanwhile, boasted some 63 million pounds ($81.5 million, 73 million euros) of new players in Mane, Georginio Wijnaldum and Ragnar Klavan.
But after some untidy opening exchanges, Arsenal were first to impose themselves on the game.
Aaron Ramsey, playing in a number 10 role, had a couple of sights of goal, jabbing a shot straight at visiting goalkeeper Mignolet and then lifting an ambitious lob well over the bar.
Liverpool left-back Alberto Moreno was proving a typically chaotic presence and on the half-hour his untidy, full-blooded challenge on Walcott gifted Arsenal a penalty.
Walcott took the spot-kick himself and saw Mignolet plunge to his right to push it away.
- Mane's Klopp piggyback -
But barely a minute later the England forward atoned as Alex Iwobi's pass caught Moreno out of position and Walcott clipped a low shot inside the left-hand post.
Stung, Liverpool responded, Wijnaldum seeing a side-foot effort saved by Petr Cech, and in first-half stoppage time Coutinho netted the equaliser in memorable fashion.
Having won a free-kick when going down a little easily under pressure from Holding, the Brazilian took charge of the set-piece himself, curling an exquisite 25-yard shot right into the top-left corner.
It had been an evenly balanced first half, but within 18 minutes of kick-off in the second Liverpool had the game in the bag thanks to three delightful goals.
Four minutes in, Coutinho sent Wijnaldum scampering towards the byline on the left with a stabbed first-time pass.
The former Newcastle United midfielder cut inside and lifted a cross to the back post, where Lallana chested the ball down before steering it past the outrushing Mignolet.
Seven minutes later, an attack from the opposite flank produced the same result, Nathanie Clyne crossing from the right and Coutinho darting in at the near post to volley home.
Mane opened his Liverpool account in the 63rd minute with a splendid goal, driving down the right and slipping between Chambers and Nacho Monreal before hammering a left-foot shot past Mignolet.
The Senegal international celebrated by leaping onto Klopp's back. The home fans booed.
A minute later, substitute Oxlade-Chamberlain gave Arsenal hope by cutting in from the left and beating Cech via a deflection off Dejan Lovren.
Wenger gave new signing Granit Xhaka his debut as a replacement for Mohamed Elneny, while Coutinho was forced off by cramp, with Emre Can coming on.
Chambers further reduced the arrears with a 75th-minute header from Santi Cazorla's free-kick, but Liverpool held on to secure a breathless win.