While the former champions had a comfortable 3-1 victory at Benfica, Manchester City had a 1-0 win over Atletico Madrid.
Ibrahima Konate and Sadio Mane gave Liverpool a 2-0 half-time lead in Lisbon, while Darwin Nunez pulled one back.
But a late Luis Diaz strike means that Jürgen Klopp’s team are heavy favourites to wrap up the quarter-final tie at home next week.
Before that comes on Sunday an English Premier League (EPL) table-topper at Manchester City who got their win on the night from Kevin de Bruyne.
The Belgian scored in the 70th minute against Atletico Madrid who had ousted Manchester United in the previous round and can still reverse matters at home on April 13.
The other two first leg matches in the Champions League are billed for Wednesday, with holders Chelsea versus record 13-time winners Real Madrid and Villarreal versus Bayern Munich.
As expected, last year’s runners-up Manchester City dominated possession at the Etihad but lacked ideas against Atletico Madrid who defended deep with a five-man backline.
The game came to life after the break when Ilkay Gündogan had a curling shot deflected and Aymeric Laporte headed inches high in the 62nd minute.
De Bruyne finally broke the deadlock with 20 minutes left for Pep Guardiola’s side with a low shot into the far left corner off a pass from the lively substitute Phil Foden.
Stefan Savic denied De Bruyne a second 10 minutes later by clearing his shot in front of the goalline.
It left the team of Diego Simeone —— who left without a handshake with Guardiola —— still well-placed to turn around matters at the Wanda Metropolitano.
“Phil found me perfectly, I just needed to stay composed and luckily I did,” De Bruyne later said of his goal.
“It was a very hard game. The first half was tight. In the second half we had a couple of chances and it was good that we took one.
“We have to stay composed. 1-0 is 1-0. I expect a similar game over there. If the game is tight they will have to attack a little bit more.”
In Lisbon, Konate headed Liverpool 1-0 up in the 17th minute off Andrew Robertson’s corner-kick.
Team captain Nicolas Otamendi nodded wide on Benfica’s best chance.
They were punished seconds later when Mane made it 2-0 off Diaz heading on Trent Alexander-Arnold’s long cross-field ball in the 34th minute.
Benfica were back in the game in the 49th minute after Konate failed to clear a cross and Nunez behind him happily accepted this gift.
But Diaz ended all doubts in the 87th minute, slotting into the empty net after rounding goalkeeper Odysseas Vlachodimos to put the Reds firmly in command as they head to Anfield.
“It was a tough game, (we were) on top in the first half and could have gone in a couple more up.
“Then we wanted to keep it tight but they got a goal which lifted the crowd and we got a bit sloppy. The third goal was big,” Robertson said.