Alexis Mac Allister's opened the scoring in the 52nd minute and Cody Gakpo headed the second in the 77th for Liverpool's first win over Real since 2009, according to German news agency (dpa).
Liverpool goalkeeper Caoimhin Kelleher, the deputy of injured Alisson Becker, saved a penalty from Kylian Mbappe in the 61st, and Mohamed Salah fired wide from the spot at the other end in the 70th.
Salah's miss didn't matter in the end as Liverpool are the only team with a maximum 15 points from five matches, and the runaway Premier League leaders have boosted morale for the next big Anfield game on Sunday against struggling domestic champions Manchester City.
Real languish in 24th place which is the last spot for a play-off berth.
Inter Milan are behind Liverpool on 13 and Barcelona and Borussia Dortmund have 12 each.
Last season's finalists Dortmund won 3-0 at Dinamo Zagreb from Jamie Gittens, Ramy Besebaini and fit again Serhou Guirassy.
Ten-man Monaco failed to join Inter on 13 points when they fell 3-2 against Benfica for their first defeat, with veteran Angel di Maria setting up Arthur Cabral in the 84th and Zeki Amdouni in the 88th.
Elsewhere, PSV Eindhoven's Malik Tillman scored in the 87th and 90th minute, and Ricardo Pepi in the fifth minute of stoppage time for an extraordinary 3-2 comeback victory against 10-man Shakhtar Donetsk.
Aston Villa had a last gasp goal from Morgan Rogers disallowed in a 0-0 draw against Juventus, Celtic drew 1-1 with Club Brugge and while finally scoring their first goal of the campaign, Bologna lost 2-1 against Lille.
In the early fixtures, Red Star Belgrade thrashed VfB Stuttgart 5-1 to earn their first points, and so did Sturm Graz who edged Girona 1-0.
-- BERNAMA
