Mohamed Salah scored twice to follow up goals from Roberto Firmino, in the fourth minute, and Philippe Coutinho as Liverpool took a 4-0 lead at halftime.
Firmino added a second goal before Alex Oxlade-Chamberlain — his first for Liverpool — and Trent Alexander-Arnold completed the rout for Maribor's heaviest loss in the competition.
"Our attitude was outstanding — we were spot on from the first second," Liverpool manager Juergen Klopp said.
Liverpool started off with draws against Sevilla and Spartak Moscow, but this win lifted the English team to the top of the group on goal difference from Spartak, which thrashed Sevilla 5-1.
Liverpool has struggled to turn chances into goals in recent games, notably draws in the English Premier League against Burnley and Manchester United that have dropped Klopp's team off the title pace.
"We had bad luck in the last three or four games," Salah said. "Everyone was confident a good result was coming."
It was the biggest margin of victory for an English club away from home in the European Cup or Champions League.
Two other teams have also won 7-0 in the Champions League: Marseille over MSK Zilina in 2010 and Shakhtar Donetsk over BATE Borisov in 2014.
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