Curtis Jones struck the woodwork in the opening 10 minutes with a fine strike from the edge of the box, but despite a succession of sights of goal for both teams the match remained goalless at the break.
The match remained extremely open after the break, with Davy Klaassen missing the target with a header and David Neres hitting the post, but it was Jones who opened the scoring with a fine finish from a narrow angle.
Roberto Firmino had a late shot tipped onto the post by Andre Onana late on, before Caoimhin Kelleher made an even later stop to deny Klass-Jan Huntelaar an equaliser.
Here are the player ratings from the clash at Anfield.
Caoimhin Kelleher - 7. A comfortable early save or two settled any nerves and he put in a largely steady showing on his European debut - then made a great late save to top it off.
Neco Williams - 6. Delivered the cross for the goal, though at the other end was beaten several times.
Joel Matip - 7. Routine work for the most part although did lose Klaassen several times on crosses.
Fabinho - 9. Several important blocks and interceptions from the stand-in centre-back.
Andy Robertson - 6. Fairly quiet considering his usual importance. Did take an early knock to the ankle.
Jordan Henderson - 7. A good showing on the ball and hard-working defensively.
Curtis Jones - 8. Hit the post and had another early shot saved. Often the link man between midfield and attack. Deserved first European strike for the midfielder.
Gini Wijnaldum - 7. Standard performance from the No. 5: reliable, won back the ball, protected the back line and covered every inch of grass.
Mohamed Salah - 7. Made lots of brilliant runs which weren’t spotted. Wasted one or two openings when he might have taken on the shot earlier.
Diogo Jota - 5. Saw plenty of the ball but his final pass and timing of the ball was hopelessly lacking.
Sadio Mane - 7. Liverpool’s biggest threat running behind the defence but again failed to unduly trouble the goalkeeper.
Andre Onana - 7. Made a good save or two but made the crucial error when he misjudged a cross, leading to Jones’ goal. Brilliant late save to deny sub Firmino.
Noussair Mazraoui - 7. Did well against Mane in the channel. Had a great early effort saved.
Perr Schuurs - 7. Covering defensive runs were good and a couple of tackles were key, though he was beaten one-on-one more than once twice.
Daley Blind - 6. Struggled with the movement of the Liverpool front men. Booked too.
Nicolas Tagliafico - 6. Didn’t get forward on the overlap as much as he would usually like to and couldn’t track Salah all the time.
Davy Klaassen - 7. Made so many great runs forward but missed the target two or three times when he should have troubled the keeper.
Edson Alvarez - 5. Sat back early on before pushing on into midfield more but had little impact on the ball and couldn’t stop the home team’s counters.
Ryan Gravenberch - 7. Clever movement, always looked forwards for his first pass.
David Neres - 7. Some excellent crosses for his team-mates to attack. Taken off injured in the second half.
Antony - 7. Started well and had a couple of good dribbles infield for shots, but couldn’t trouble Kelleher.
Dusan Tadic - 7. Caused lots of problems with his movement into deep areas and clever crosses.