Liverpool have too much in the tank for Manchester United
Klopp's side survived an early scare when Martial converted a penalty, to draw at Old Trafford and go through thanks to their 2-0 win in the first leg.
Liverpool booked their place in the Europa League quarter-finals as a fabulous goal from Coutinho helped the Reds knock out Premier League rivals Manchester United at Old Trafford.
With Louis van Gaal's home side trailing 2-0 from last week's first leg at Anfield, Anthony Martial won and converted a first-half penalty to bring United back into the reckoning.
However, Coutinho conjured up a magnificent solo effort to leave the hosts needing to score three more, outpacing right-back Guillermo Varela before dinking a cool finish past goalkeeper David de Gea.
With United unable to add to Martial's spot-kick, Jürgen Klopp's charges sealed a 3-1 aggregate victory.
Here's how it all unfolded, starting with the night's starting line-ups:
Man Utd: De Gea, Varela, Smalling, Blind, Rojo, Carrick, Fellaini, Lingard, Mata, Martial, Rashford.
Liverpool: Mignolet, Clyne, Lovren, Sakho, Milner, Can, Henderson, Lallana, Firmino, Coutinho, Sturridge.
Five minutes to go to kick-off, and the players are lining up in the tunnel. Manchester United looking very fetching in their anthem jackets (formerly known as tracksuit tops).
And the teams are out! The Europa League anthem is almost as spine-tingling as its Champions League counterpart. Not.
Peeeep! They're off!
10' | As expected it's a fantastic atmosphere at Old Trafford, where it's a cagey start. A period of sustained possession for Liverpool, greeted by whistles from the home faithful, then sees the hosts get their foot on the ball for the first time, but as yet there's little in the way of attacking threat from either team.
12' | Let's be honest, this hasn't been a footballing advert for the Premier League so far.
15' | Juan Mata almost gets in Marcus Rashford running into the box, but Liverpool defender Nathaniel Clyne is alive to it. Still waiting for our first clear opening at the Theatre of Dreams.
18' | Mignolet saves! Marcos Rojo hangs it into the box, makeshift left-back James Milner just stands and stares - and Jesse Lingard heads firmly towards the corner, but the Belgian is there!
19' | You wait 20 minutes for a goalmouth opening, and two come along! Hot on the heels of Lingard's header, Mata scuffs a presentable shooting chance across goal inside the area.
22' | United growing in confidence now. Anthony Martial receives on the left, cuts inside and fires towards goal, but it's high. Encouraging signs for the hosts, though.
24' | Sturridge looks to bite back but his shot is blocked, before Rashford flops down in the Liverpool box to meek penalty shouts - but that was never going to be given.
25' | Lovren feeds Coutinho approaching the box, and had the Brazilian's control been on the money that would have been a great shooting opportunity. But his first touch is long, and the chance is lost.
29' | Great stop from De Gea! Lallana is breaking forward on the right and looks to cross low, his ball deflected into the path of Coutinho, who meets it with a sweet, low strike - but the Spaniard is quickly down to get a hand to it. Top keeping.
30' | Now Firmino with a low drive, which is gathered by De Gea.
31' | Penalty for United! Martial looks to go past Clyne, the England man sticks out a boot and the Frenchman falls...
32' | Gooooal! Martial picks himself up and sends Mignolet the wrong way from the spot! Is the comeback on? Man Utd 1-0 Liverpool.
34' | Sturridge earns a dangerous free-kick on the right edge of the United box. Boos ring out as he prepares to deliver it... And it's off the bar!
35' | This has grown into a very watchable game.
37' | Now Lingard falls for United on the far side of the Liverpool box, and the hosts have a mirror-image of a Liverpool's free-kick - but theirs produces little threat and is cleared.
38' | Rashford gets to the byline, and has United bodies waiting by the six-yard box to turn the pull-back into the net, but Liverpool clear. I take back my comment of 12'.
41' | Well well well, what a chance for Jordan Henderson. The Liverpool captain feeds Coutinho and advances into the area to receive the Brazilian's wonderful reverse return ball, but steers it over the bar. Klopp has his hands on his head, and well he might.
43' | As John Motson would say, this is a ding-dong tie. Fellaini reaches a dropping ball on the byline to find Rojo free eight yards out, and the Argentine puts it wide! In the space of a minute we've been close to tie almost over, then on the cusp of 2-2.
45' | What. A. Goal. Coutinho picks it up on the left, bursts by full-back Varela and impudently dinks it into the net past the flailing arms of De Gea at the near post. Stunning. Louis van Gaal's men need three more now. Adiós United? 1-1 (1-3 agg.)
HT: Manchester Utd 1-1 Liverpool. It took time to get going at Old Trafford, but once it did, what a fantastic game we've been treated to so far. There may have been a sense of regret before the tie that the sides' first ever continental meeting was to be in Europe's second-tier tournament; however, in the words of our Spanish commentary team: "Whatever the competition, a clash between these two arch rivals is always a sight to behold."
Peeep! The second 45 is underway at Old Trafford. There's a half-time change for Man Utd, as Varela - for whom being skinned by Coutinho proves to be his last contribution to tonight's proceedings - is replaced by Antonio Valencia.
49' | Fellaini's wild elbows earn him a booking - one that means he'll miss United's next European game. Should he have even seen red?
51' | Now Clyne drags down Rashford right on the edge of the left-hand side of the area, and goes into the book. Dangerous free-kick, but Carrick's near-post delivery is turned out for a corner, which comes to zilch.
56' | Blind hesitates and Sturridge has the chance to run at an exposed United defence midway inside the home side's half, but the Dutchman recovers to snuff out the attack. The Red Devils are then back on the attack - they're not giving this up for lost, not by a long chalk - with Sakho first sliding in to prevent Rashford from shooting, before a renewed push from Van Gaal's outfit sees Rashford hit it narrowly wide from the edge of the box.
60' | Now Fellaini has a headed chance, but nods straight at Mignolet. United are pushing to get themselves back into this tie - and a goal in the next few minutes would certainly put the proverbial cat among the pidgeons.
62' | Lovely stuff from Coutinho as he produces a silky stepover and glides by three men, before feeding Sturridge on the right, but the England international slices wide. He really should have made more of that. Matteo Darmian replaces Marcos Rojo for United.
63' | Martial plays a neat one-two with Lingard, but the ex-Monaco man fires over.
67' | Martial and Mata link up nicely around the edge of the box, but as soon as it comes to Fellaini the Belgian lacks his team-mates' precision, dallies and a promising attack is dead. The home support voice their disapproval.
68' | First change for Liverpool as Origi comes on for Sturridge, who doesn't look best chuffed at being hooked.
71' | Third sub for United, Carrick making way for Schweinsteiger, while Liverpool boss Klopp also shuffles his pack by withdrawing skipper Henderson and bringing on Allen.
72' | Wasteful from Fellaini, the Belgian sclaffing it over after good work from Rashford to cut it back for the big man. "Fellaini isn't winning many admirers tonight," says our Spanish co-commentator.
75' | Ooooooh... What a goal that would have been. Emre Can advances into the left-hand side of the box and demonstrates wonderful vision to backheel to Coutinho, who drops a shoulder and cuts inside Smalling, but his shot on goal is gathered by De Gea.
78' | Ufff! Another chance for Liverpool as Coutinho gets it out of his feet on the edge of the 18-yard area and pings it towards goal for custodian De Gea to palm it over.
79' | Sub Schweinsteiger earns a deserved yellow for scything down Lallana just short of the area. Coutinho over the free-kick... But it flies over.
82' | The TV coverage cuts to a shot of home fans making their way out of Old Trafford. United's forward threat has plummeted in the last ten minutes or so, and the Merseysiders are comfortably shepherding this tie to its conclusion.
85' | Firmino - who was booked a couple of minutes ago for a challenge on Lingard - is replaced by Benteke.
87' | Schweinsteiger and Blind have tame efforts blocked, before Lingard fires wide to cheers from the Liverpool fans - with three minutes remaining, it's party time among the travelling support.
90' | Cries of "go on!" from the away section as Benteke runs at the Man Utd defence before digging out a low shot that De Gea is equal to, before the former Atlético keeper then has to be alert to parry Origi's effort.
FT: Manchester United 1-1 Liverpool (Liverpool win 3-1 on aggregate)
That's it at Old Trafford, where Liverpool did as manager Jürgen Klopp had urged them to pre-match, and finished the job. Anthony Martial's penalty had offered Manchester United hope of producing a turnaround, but Coutinho's wonder goal on the stroke of half time simply proved too heavy a blow for the hosts to come back from. United came out after the break in no mood to give up the ghost, but ultimately were second best to the men from Merseyside over the two legs - and Klopp's boys fully deserve their last-eight berth.