Liverpool's Brazilian playmaker danced down the touchline and glided past a defender, before chipping the ball over keeper David De Gea from a tight angle to quieten the home crowd, who were sensing a comeback after Anthony Martial's penalty opener.
Juergen Klopp's side, who did most of the damage in a 2-0 first-leg victory, will be joined in the last eight by the German coach's former club Borussia Dortmund, who brushed aside Tottenham Hotspur 5-1 over two legs, after a 2-1 win in London.
The quarter-final lineup was completed by holders Sevilla, Athletic Bilbao, Shakhtar Donetsk, Braga, Sparta Prague and Villarreal with the draw taking place on Friday.
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Liverpool's rivalry with United stretches back over 122 years, but the two sides were playing out their bitter feud on the European stage for the first time in the two-legged tie.
The 196th competitive encounter between the clubs had the potential to be an epic with United needing to come out all guns blazing in order to have a chance.
When Martial was tripped by Nathaniel Clyne with barely half an hour gone and picked himself up to coolly dispatch the penalty the comeback looked on.
United had already squandered a number of chances and Liverpool looked vulnerable, but any momentum the home side had built up was crushed a minute before halftime when Coutinho's goal left them needing to score four.
The second half was an anti-climax as United looked bereft of belief and Liverpool threatened on the break.
Liverpool are England's only surviving representative in Europe's second-tier competition as Tottenham's Europa League adventure was ended emphatically by Dortmund.
Led by Pierre-Emerick Aubameyang, who scored twice on Thursday to add to the goal he scored in the first leg, Dortmund proved too strong for the Premier League outfit, who left top scorer Harry Kane on the bench with an eye on their domestic title challenge.
Aubameyang opened the scoring with a superb dipping effort after 24 minutes and doubled the advantage with a venomous effort 19 minutes time.
Son Heung-Min's consolation for Spurs could not stop them falling to their first defeat in 10 European home games.
EIGHT MEN
Sevilla, chasing a record third straight title, were out of sight by halftime in their second leg clash against Basel as two Kevin Gameiro strikes in the space of two minutes and a goal for Adil Rami completed a 3-0 victory after the first leg ended 0-0.
Fenerbahce ended with eight men as they gave up a 1-0 first-leg advantage in a 4-1 defeat to Braga, who ended the Turkish side's involvement with goals from Ahmed Hassan, a Josue penalty and further efforts from Nikola Stojiljkovic and Rafa Silva.
Mehmet Topal, Alper Potuk and Volkan Sen, who received two yellow cards in stoppage time, were all dismissed for Fenerbahce.
Aritz Aduriz scored late in the second half to send Athletic Bilbao into the quarter-finals on away goals, despite a 2-1 defeat to Spanish rivals Valencia.
Valencia boss Gary Neville has had little to celebrate since joining the club in December but looked on course for some rare joy when his side scored twice in the first half through Santi Mina and Brazilian defender Aderlan Santos.
Elation turned to disappointment, however, when Aduriz poked home his eighth Europa League goal this season after 76 minutes following a brilliant flick from Raul Garcia, giving Bilbao a crucial away goal to add to their 1-0 win in the first leg.
There was no late drama in Italy where Lazio were taken apart by Sparta Prague, who scored three times in the first half to complete a 3-0 away win and 4-1 aggregate victory.
Villarreal went through 2-0 on aggregate after a goalless draw at Bayer Leverkusen while Shakhtar Donetsk's Croatian striker Eduardo scored in stoppage time for a 1-0 win over at Anderlecht to complete a 4-1 aggregate victory.
(Editing by Martyn Herman)
