The Reds had beaten Maribor 7-0 in the reverse game a fortnight ago but Jurgen Klopp's men found the Slovenian champions much harder to open up at Anfield.
Liverpool had taken just four minutes to break the deadlock in Slovenia but they were made to wait until the second half after a frustrating first 45 minutes that saw no clear-cut chances at either end.
Mohamed Salah's delightful flick from a whipped right-wing Trent Alexander-Arnold cross eased the tension, although Jasmin Handanovic soon stopped James Milner from extending the lead by saving the midfielder's penalty after Roberto Firmino was brought down.
Milner somewhat made amends with a lovely assist for the second goal after 64 minutes, though, Emre Can bending in a fantastic first-time strike that gave Handanovic no chance.
Sevilla's victory over Spartak Moscow leaves Liverpool - who wrapped up the win through Daniel Sturridge - top with eight points ahead of their matchday five meeting against the LaLiga side, while bottom side Maribor can mathematically still qualify, but are four points adrift.
Alex Oxlade-Chamberlain was handed a rare start in the absence of injured Philippe Coutinho and the England man had the first opening, seeing a shot blocked by Jean-Claude Billong.
Jurgen Klopp used the chance to rest Jordan Henderson but the club captain had to come on after just 15 minutes when Georginio Wijnaldum succumbed to injury.
Liverpool finally forced a save from Handanovic after 29 minutes, the goalkeeper at full stretch to tip over a deflected Firmino header that was looping into the top-left corner.
Can and Salah wasted decent chances as Maribor held firm throughout the first half despite Liverpool's dominance of the ball, their share of the possession topping 80 per cent.
Maribor's gritty resistance was broken a mere three minutes after the restart, however.
13 - Mohamed Salah has been directly involved in 13 goals for Liverpool this season (10 goals, 3 assists); more than any other player. Vital pic.twitter.com/e38Ny3u2fB— OptaJoe (@OptaJoe) November 1, 2017
Joel Matip fed Alexander-Arnold and the young full-back provided a fine whipped cross that Salah turned in with a neat improvised first-time finish for his fourth Champions League goal of the season.
Firmino drew a penalty by being fouled by Aleksandar Rajcevic after beating the Maribor defender with a fine piece of skill, but Handanovic brilliantly denied the usually reliable Milner from 12 yards by diving low to his left to push the shot on to the post.
Salah should have buried a header from a fine Alberto Moreno cross after 55 minutes as Liverpool threatened to run riot, before Billing headed a corner wide from a rare Maribor attack.
But the second goal duly arrived thanks to Liverpool's most incisive move, Can swapping passes with Milner and placing a terrific 20-yard shot into Handanovic's bottom-left corner.
Liverpool added to their lead in the last minute of normal time, substitute Sturridge lashing in a left-footed strike after Maribor failed to clear a Moreno centre, but a November 21 trip to Seville will remain decisive for Klopp's side.
KEY OPTA STATS
- Liverpool have kept four clean sheets in their last five games in all competitions, after managing none in the seven games before that.
- Maribor are without a win in their last 15 Champions League games (D5 L10).
- Liverpool have failed to score with five of their last seven penalties in all competitions, including each of their last three.
- Emre Can netted his first-ever Champions League goal, in his 13th appearance in the competition.
- Daniel Sturridge bagged his first Champions League goal in almost seven years, since scoring for Chelsea against MSK Zilina on November 23rd 2010.