First-half goals from Mohamed Salah, Alex Oxlade-Chamberlain and Sadio Mane means City have it all to do in the return match.
The Blues came into the game in excellent form and started the match confidently, but the concession of three goals in under 20 first-half minutes stunned Pep Guardiola's men.
The verve and fluidity that has characterised so much of City's play this season was frustratingly elusive early on against a committed and capable Liverpool side.
City have often enjoyed more than 70 per cent of the ball in matches this season but that was not the case here in the first half, with the play often broken and open, robbing the Blues of the control the players crave.
Guardiola's side were much improved after the break but in the end - just as was the case in the Premier League defeat here in January - three Liverpool goals in quick succession did the damage and City must pull off a remarkable comeback in order to progress.
Still, City have won heavily on numerous occasions at the Etihad this season, including against Liverpool, so this tie is by no means over.
City were behind after Liverpool pounced on a loose ball to break quickly into the Blues' box. Nicolas Otamendi's challenge held up Firmino's progress, but Kyle Walker could not clear an awkwardly bouncing ball, and the Brazilian stabbed a cross through for Salah to finish.
Guardiola's men tried to counter and Sane, prompted by David Silva, drove forward only to drag his shot wide.
Moments later City were two down when a half-clearance made its way to Oxlade-Chamberlain who unleashed an unstoppable shot from 20 yards.
Liverpool had the momentum, and soon a third goal.
Salah's shot rebounded back to him on the edge of the box and he curled a cross in for Mane to nod home.
City started the second half brightly, and when Fernandinho picked out Sane with a wonderfully disguised pass his cross flashed agonisingly across the six-yard box with no-one able to finish.
City were certainly enjoying more joy down the left-hand side as the game wore on, with Sane a constant outlet and Kevin De Bruyne and Silva probing, but the chances didn't fall. Gabriel Jesus did have the ball in the net late on but Sane had earlier been flagged offside in an extremely close call.
In the end the Blues could not find the breakthrough. City have just one win in 37 years at this ground and the Anfield hoodoo continues.
Matchday App Star Man: Leroy Sane
A willing runner all night and he pressed the Liverpool defence back on numerous occasions in the second half.
Key moment - Salah goal
City had started the game strongly but the early Liverpool goal flipped the momentum for that key 20-minute spell.
Guardiola opted to bring Gundogan in for Sterling in a 4-2-3-1 formation. De Bruyne and Fernandinho sat deeper, with Leroy Sane and Gundogan on the left and right respectively, either side of David Silva. With Sergio Aguero still out injured, Gabriel Jesus led the line.
City face United in the Manchester derby on Saturday (kick-off 5.30pm) - a game in which the Blues could clinch the Premier League title. Following swiftly on from that encounter is the second leg against Liverpool, at the Etihad Stadium on Tuesday night (kick-off 7.45pm).