City made the perfect start as they tried to reel in a 3-0 deficit from the first leg when Raheem Sterling set up Gabriel Jesus for a cool finish after only two minutes - but they never recovered from a contentious incident just before half-time that saw manager Pep Guardiola sent off.
Bernardo Silva had struck the post and with Liverpool under intense pressure, Leroy Sane scrambled home what City thought was a crucial second.
The goal was ruled out for offside even though the ball came off Liverpool's James Milner and Guardiola marched on to the pitch in an explosion of fury to remonstrate with referee Antonio Mateu Lahoz. The outburst saw him forced to watch the second half from the stands.
With Guardiola away from his technical area, City lost momentum and the visitors delivered the killer blow of the away goal when, inevitably, Mohamed Salah chipped in his 39th goal of the season after a powerful run by Sadio Mane in the 56th minute.
Roberto Firmino's classy finish 13 minutes from time completed the formalities and Liverpool will now have their sights set on a sixth triumph in Europe's elite competition.
The Reds will discover their semi-final opponents when the draw is made on Friday at 12:00 BST in Nyon, Switzerland.
Magic Salah delivers again
Salah's fitness was the major topic of conversation around Liverpool's team selection before this game - and his decisive contribution in the 56th minute demonstrated exactly why.
The striker was substituted in the first leg with a groin problem but was fully fit and restored at the start as Liverpool protected that three-goal lead from Anfield.
The Egyptian was powerless in the first half as City tried to claw that deficit back in a thunderous first 45 minutes that saw Jurgen Klopp's side under siege.
In this golden season, however, Salah only needs one chance and so it proved again as he collected the ball as it ran loose from Mane's drive into the area before delivering a composed lofted finish with the confidence of a man who feels he cannot miss.
These are the sort of game-changing contributions needed at this elite level and Salah delivers them.
It was fitting that the underrated Firmino added Liverpool's second with equal calm as City ran out of hope - but it was Salah's cool head in this frantic atmosphere that changed the course and emphasis of this quarter-final second leg.
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