Capping off a stunning team move for the opening goal on 15 minutes, Mane scored his first goal for the club as they took a deserved lead in their International Champions Cup clash.
The wide-man later helped double the lead as former Liverpool midfielder Javier Mascherano inadvertently turned into his own net from the winger's cross.
Half-time substitute Divock Origi added a third only a few minutes later, while Marko Grujic's looping header late into injury time put the finishing touches on a memorable evening.
Fast-paced start blows star-studded Barcelona away
They faced a strong Barca starting eleven with Lionel Messi and Luis Suarez - reuniting with Liverpool for the first time since a £75 million move to Spain two years ago - among those to feature from the outset.
89,945 turned out at the national stadium to watch two European heavyweights go head-to-head - the second-highest attendance ever since the new Wembley re-opened in 2007.
And they were rewarded with an enthralling 90 minutes of football, Mane taking less than 30 seconds to demonstrate his abilities as Liverpool came flying out of the blocks.
Mane exchanged a one-two with Adam Lallana to beat his man down the right-hand side and his cross reached Georginio Wijnaldum inside the area, but Sergio Busquets was on hand to close the midfielder down to deflect his shot wide.
It wasn't long before the home side forced the breakthrough. Lallana won possession from Juan Camara inside Barcelona's half, forcing a quick break that led to neat passing between Firmino and Lallana.
The Englishman slipped a pass through the legs of Mascherano into Mane, who needed just one touch before firing high past Marc-Andre ter Stegen to send the Liverpool supporters inside Wembley into raptures.
Klopp's side soon found themselves under pressure as Luis Enrique's men sprang into life, with Messi going close as he struck the far post from close-range after Suarez's set-up.
Shortly after, Simon Mignolet denied Suarez from the edge of the box before Arda Turan dragged the Catalans' follow-up attempt wide.
Mignolet was forced into an excellent save at point-blank range in the 40th minute after winger Munir El Haddadi's crossfield ball found Suarez, but the Belgian 'keeper was quick to narrow the angle and block the Uruguayan's half-volley.
Liverpool rode the storm to protect their lead into half-time and another fast start in the second-half rewarded them with two goals to extend their cushion.
The only blot on the copybook was an injury to Milner, who had to be brought off for Alberto Moreno shortly before half-time and went straight down the tunnel with a muscle injury.
Quickfire second-half strikes extend Reds' lead
Yet again, more superb pressing allowed Mane to pickpocket Mathieu down the right and after exchanging passes with Origi - his low cross into Henderson in the six-yard box forced Mascherano to put past his own goalkeeper.
No sooner was it 2-0 than it was 3-0 as yet more harrying forced Barcelona into mistakes. Kevin Stewart, having also been introduced at the break, robbed the ball from Busquets and set his teammates on their way.
His first-time forward ball released Origi through on goal and the Belgian's pace kept him ahead of Jeremy Mathieu and Mascherano before he slipped an effort beneath substitute goalkeeper Claudio Bravo.
It should have been a four-goal advantage by just the 52nd minute when Stewart fed a through ball for Firmino, but with the goal gaping he could only fire his attempt across goal wide.
Enrique opted to make a whole host of changes on the hour, introducing Gerard Pique, Andres Iniesta, Ivan Rakitic and summer signing Lucas Digne - but they had little positive impact.
Instead it was Liverpool who looked the more dangerous team, and the more likely of adding to the scoresheet, with Mane close to completing a fine solo goal in the 71st minute.
The Reds' No.19 wriggled past a number of challenges as he drove his way into the penalty area, but after cutting on to his left foot - he could only curl wide of Bravo's far post.
Mass changes for both sides - Mane, Wijnaldum and Firmino going off for Danny Ings, Lazar Markovic and Grujic in Klopp's substitutions - altered the tempo as the game wore on.
Grujic header seals resounding victory
Suarez was withdrawn to a rapturous reception from the Liverpool supporters as they continued to cruise towards victory in the sun-drenched capital.
But Grujic rounded off the scoring late on, Henderson's strong challenge on Iniesta on the halfway line allowing Origi to start a break.
Fortunately, though he failed to beat his marker, the ball fell kindly for Markovic who chipped a ball over the back-line and into the path of Grujic who leapt high to direct a header which hung up in the air before dropping beyond Bravo for a fourth.
That added even extra gloss to an unforgettable evening for Klopp's side as they recorded an emphatic victory over the reigning La Liga champions.
The Reds play their final fixture less than 24 hours after facing Barcelona, with a much-changed side likely to be named for the trip to Germany to take on 1. FSV Mainz 05 in their last warm-up.