Liverpool's strong start under Arne Slot continued as Darwin Nunez's goal extended their 100% winning record in the Champions League.
Nunez marked his fourth start under Slot with a 27th-minute finish from close range against RB Leipzig as the Reds made it 11 wins out of 12 in all competitions for their Dutch manager.
Uruguay forward Nunez turned in a fine shift. As well as touching home Mohamed Salah's header for the decisive moment of the game, he went close to a second goal before half-time when his header was tipped over the crossbar by former Liverpool reserve goalkeeper Peter Gulacsi.
Gulacsi also denied Virgil van Dijk after the defender was left unmarked at a corner.
Liverpool produced a controlled performance in Germany and looked to have won a penalty when Nunez, who was a real menace, was clipped by Willi Orban in the 38th minute only for the officials to wave away the visiting team's protests.
There were one or two anxious moments for the Reds before Liverpool were able to celebrate an eighth straight win in all competitions.
At 0-0, goalkeeper Caoimhin Kelleher had been almost embarrassed when a weak headed clearance after charging off his line led to a chance for Benjamin Sesko whose curled attempt dropped just wide.
Kelleher made amends with smart saves to keep out Sesko and Xavi Simons.
RB Leipzig also had two goals ruled out for offside, but Liverpool showed good game management to get over the line and leave themselves in a strong position before Champions League home games against Bayer Leverkusen - managed by former Reds midfielder Xabi Alonso - and record 15-time winners Real Madrid.
This win means Liverpool join Premier League rivals Aston Villa with three wins out of three in European football's top club competition.
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