Having drawn their opening two games, Liverpool made a dream start when Mohamed Salah burst into the box before squaring for Brazilian forward Roberto Firmino to stroke home in the fourth minute.
Nine minutes later Firmino's compatriot Philippe Coutinho converted James Milner's smart pullback to double the lead before Salah finished smartly at the near post as Liverpool ran riot in the opening 19 minutes.
Liverpool's fluid, dynamic front three continued to cut loose and the effervescent Salah capped a stunning first half for the visitors, converting Alberto Moreno's low cross to make it 4-0 in the 39th minute.
Halftime did little to halt their momentum and Firmino added a well-deserved second finishing from Coutinho's fierce cross with a deft header in the 54th minute.
Substitute Alex Oxlade-Chamberlain, with his first Liverpool goal, and Trent Alexander-Arnold completed the rout in the closing stages as the visitors registered their biggest win under manager Juergen Klopp.
Liverpool have five points after three games to lead Group E from Spartak Moscow on goal difference, while Spanish side Sevilla sit one point adrift in third.
Maribor have one point.
(Writing by Tom Hayward; Editing by Toby Davis)
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