Jurgen Klopp’s side worked tirelessly as a group to put pressure on City and scored three goals in the first half to take command of the first-leg tie.
Mohamed Salah broke the deadlock after just 12 minutes and Alex Oxlade-Chamberlain scored a long-range stunner on 20 minutes, before Sadio Mane sealed the win 11 minutes later.
A brisk start to the match saw both teams show plenty of intent, but the hosts settled quicker as they knocked City out of their flowing style with heavy pressing.
The Premier League leaders were first to threaten on six minutes when Leroy Sane collected a pass from Silva on the left and surged into the area before seeing his effort deflected into the side-netting from a tight angle,
But a defensive mistake from the visitors resulted in the opener on 12 minutes as Salah was sent racing down the right flank and combined with Roberto Firmino, who squared for the Egyptian to score from six yards out after Kyle Walker failed to clear the danger at the near post.
The Reds’ night got even better midway through the first half when Oxlade-Chamberlain controlled a firm ball into his feet and lashed an unstoppable shot into the left corner of the net from 21 yards out.
They were well on top just past the half-hour mark as Mane extended their lead. Salah was the provider with a delightful dinked cross to the back post, where the Senegalese winger headed home with ease.
Firmino looked to get in on the act four minutes before half-time when he was allowed to turn on the ball and fire a left-footed shot over the crossbar from 20 yards out after not being closed down.
Virgil van Dijk then went close in stoppage time as he glanced a header wide of the right post from a free-kick on the left, with the Merseysiders deserving of their 3-0 lead at the break.
Sane was presented with an early chance in the second half after being picked out at the back post by a delightful cross from Fernandinho, but the German volleyed wide of the far post on 52 minutes.
City enjoyed more possession after the restart but they found the Reds defence impenetrable. Nicolas Otamendi managed to get on the end of a 69th-minute corner, but his header sailed well over the bar.
The only other chance that came their way was in the 75th minute when Gabriel Jesus connected with Raheem Sterling’s cross from the right, but the ball was delivered behind the Brazilian, who couldn’t direct it on target with his back towards goal.