Struggling for chances in a first-half where Georginio Wijnaldum was taken off with an injury, those that the Reds did create were kept out by veteran 'keeper Jasmin Handanovic.
Ox in and keen to impress
Klopp handed Alex Oxlade-Chamberlain just his second start for Liverpool since signing in August, the winger clearly keen to get the crowd on side as he launched into a couple of early tackles.
It looked like he might bag his second goal of the competition as he raced onto a low cross by Roberto Firmino, only for defender Jean Claude Billong to nip in ahead of him and clear for a corner.
Liverpool's best chance of the half would go to Firmino, whose header from an Oxlade-Chamberlain cross was kept out by another top save by Handanovic, the 39-year-old tipping the ball onto the bar.
By that point the Reds had early been forced into their first change, Georginio Wijnaldum pulling up with an ankle problem, replaced by skipper Jordan Henderson.
Hosts get the job done
Things got better at the start of the second half, Liverpool taking the lead just four minutes after the restart.
Trent Alexander-Arnold, returned to the side for this one, charged down the right and delivered an excellent cross for Salah to flick across the goal and into the net.
The next piece of action was all about Roberto Firmino, with some excellent skill earning Liverpool a penalty that they'd subsequently miss.
Aleksandar Rajcevic was to blame for Maribor, nut-megged by a lovely Firmino turn in the area, bringing the Brazilian down as he surged towards goal.
Unfortunately for Liverpool, their penalty problems continued despite regular taker James Milner returning to the spot, Handanovic getting down excellently to his left to save.
Klopp's men would extend their lead before the end of the game though, Can and then Sturridge making sure the three points were in no doubt.
Can's goal, his third of the season in this competition, was particularly nice. After patient midfield passing, the German quickened things up with a one-two via Milner, opting to rifle the ball into the bottom corner as he received it back on the edge of the area.
Chances came and went before number three went in, Sturridge striking in stoppage time to finally net a Champions League goal for Liverpool.
Alberto Moreno picked up a deserved assist at the end of another good performance, his cross not cleared by the Maribor defence as Sturridge waited patiently at the back-post, finishing smartly.
Sevilla's 2-1 win over Spartak Moscow in the other group game leaves Liverpool a chance to secure qualification by beating Sevilla in Spain in three weeks time, a result that would also leave them with a solid chance of finishing top of Group E.