Liverpool moved closer towards the Champions League knockout stages for the first time since 2008-09 with a hard-fought victory over Maribor.
Mohamed Salah broke the deadlock in the second half with a clever, flicked finish but James Milner had a penalty brilliantly saved by 39-year-old goalkeeper Jasmin Handanovic.
Midfielder Emre Can exchanged passes with Milner to curl in the second from the edge of the box and Daniel Sturridge drilled in a third late on.
Liverpool dominated the first half, holding 83% possession, but were unable to find the breakthrough - Roberto Firmino coming close with a stooping header which was tipped onto the crossbar by Handanovic.
Maribor's best chance fell to Martin Milec, but his low drive was collected by Loris Karius, the only save he made in the match.
Jurgen Klopp's side remain top and unbeaten in Group E with eight points from four games. They face Sevilla away and Spartak Moscow at home in their remaining games.
In the group's other game, Sevilla moved up to second by beating Spartak 2-1.
In their previous game against Huddersfield, Liverpool were goalless at half-time but went on to put three second-half goals past the Terriers. They did the same on this occasion.
The Anfield side had thrashed Maribor 7-0 last time out in Europe but failed to find the back of the net from 13 first-half shots.
Although Salah netted his 10th goal for Liverpool since signing from Roma in the summer, the Egyptian was wasteful in front of goal, slicing over the bar from a promising position and heading off target from just four yards out.
Liverpool also missed a penalty for the second consecutive game as Milner's effort was tipped onto the post by a full-stretch Handanovic. The Premier League side have now failed to convert their past four spot-kicks at home.
But their persistence paid off when German international Can gave his side breathing space with a well-taken finish, before England striker Sturridge came off the bench to net in back-to-back games for the first time since the start of this year.
Can said: "It was difficult in the first half because they were deep but we got more space in the second half and it was a game where we had to be patient.
"We could do better this season but it is still early. Now we are in a good position in the Champions League and hopefully we will be in the next round."
Liverpool have now lost just once in their past eight games in all competitions and a much-criticised defence has kept four clean sheets in their past five games.
Klopp will be concerned by a first-half injury to Dutchman Georginio Wijnaldum, who needs a scan on a swollen ankle sustained in an innocuous collision with a visiting player.
The midfielder often provides the shield to allow the more adventurous players to foray forward and Liverpool shipped four goals against Tottenham without him last month.
They were also held to a 1-1 draw at home to Burnley in the only other match he has missed this campaign.
This game, though, was not one where the Liverpool backline was tested by a team who have failed to win in their past 15 Champions League group games.
Such was the home side's control, centre-back Joel Matip attempted the most passes of any player on the pitch (120), closely followed by defensive partner Ragnar Klavan with 118.
Should Wijnaldum be ruled out for any length of time, he will join a growing Liverpool injury list.
Long-term absentees Adam Lallana and Nathaniel Clyne are yet to feature this season, while creative midfielder Philippe Coutinho has missed the past two games with an adductor problem.
Man of the match - Emre Can (Liverpool).
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