Mohamed Salah, Emre Can and Daniel Sturridge kept Liverpool top of Group E.
The Egyptian broke the deadlock after 48 minute with a brilliant flick from Trent Alexander-Arnold’s cross.
Then Can netted a superb second with a crisp exchange of passes with James Milner for his first Champions’ League goal.
Daniel Sturridge rounded off proceedings late on, with his first goal in the competition for seven years.
It could have been more, with Milner missing a penalty.
Georginio Wijnaldum's injury in the opening 15 minutes was the only down point for Liverpool, who wait to find out how serious the damage is.
Here are the key talking points from the routine European evening an Anfield...
1. Could Liverpool really repeat their seven goal rout?
A repeat of Liverpool’s biggest European away win was never on.
The Slovenians packed the defence, banked up the midfield and went for damage limitation.
Breaking down such a defensive wall was tough and highlight’s sparse in the first half.
The hiding dished out away from home produced a backlash, and Maribor can be proud of their effort, for their first 48 minutes, until their defence was unpicked.
2. Salah sparks Liverpool to the win
The Egyptian started up front alongside Firmino and his tenacity was needed to break the first half deadlock.
His movement, buzzing across the line, wore the defence down, and he’s essential to the Liverpool cause.
His brilliant flick opened the scoring on 48 minutes.
Salah has been directly involved in 13 goals for Liverpool this season - 10 goals and 3 assists.
3. Why do Liverpool miss so many penalties?
James Milner made it two misses in a row personally.
And Liverpool have now missed their last four penalties at Anfield in 2017.
This wasn’t a costly miss like Firmino’s against Seville.
But it’s a worry, Mo Salah failing to score from the spot against Huddersfield last weekend.
4. Has a side ever had so much possession?
Livepool dominated the ball. After 20 minutes the possession stat hit a remarkable 78pc.
By half time it was 84pc. Even Barcelona at their best struggle to match that.
It took 49 minutes for the goal to come, but Liverpool dominated.
5. So who is in charge of qualifying from Group E?
Spartak Moscow and Seville are breathing down Liverpool’s neck and they needed the banker of a home win for some leeway.
Seville prevailed against Moscow tonight and their clash next up against Liverpool could decide who win the group.