They drew 1-1 away against Olimpija Ljubljana before a 3-0 win at Anfield.
Earlier, Maribor and Liverpool did not meet.
Half chances for Firmino and Alberto Moreno followed as Liverpool looked settled against their star-struck hosts.
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Roberto Firmino and Mohamed Salah both scored twice while Philippe Coutinho, Alex Oxlade-Chamberlain and Trent Alexander-Arnold also found the net as Liverpool, with just one win in their last eight matches, ran riot in Slovenia. We need to keep looking forward. Two points from their opening two group matches was not what boss Jurgen Klopp would have expected, especially after surrendering a winning position late on at home to Sevilla in their opening game.
Maribor are making their third appearance in the Champions League after 1999/00 and 2014/15 but have never reached the knockout stages.
Liverpool face Tottenham Hotspur this Sunday in the Premier League.
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Captain for the night Milner played in place of Jordan Henderson and marshalled the Reds from start to finish, determined not to let standards slip as the game entered somewhat of a lull. Sturridge dispossessed Suler who was dawdling on the ball, before playing in his fellow countryman Oxlade-Chamberlain who sent the ball to Handanovic's left to the delight of the former Arsenal man and his manager.
Jurgen Klopp hailed Liverpool's return to form after a record-breaking 7-0 victory over Maribor in the Champions League.
Klopp was all smiles on the touchline and there was still time for teenage defender Alexander-Arnold to net in stoppage-time with a deflected long-range effort.
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Maribor and Liverpool will play the reverse fixture on November 1, the next game of the group stage, at Anfield.