Vinicius Jr struck twice either side of a Marco Asensio strike, gifted to him by Trent Alexander-Arnold, while Mohamed Salah's away goal gives the Reds hope ahead of the return leg at Anfield.
But they still have a mountain to climb after a combination of Madrid class and poor defending saw them soundly beaten in the Spanish capital.
A high line was punished by a superb pass from Toni Kroos for Vinicius' first goal, then a terrible backpass from Alexander-Arnold was seized upon by Asensio.
Salah got that all-important away goal as Liverpool improved after the break, but more sloppy defending allowed Vinicius to bag his second, with questions also hanging over Alisson, who perhaps should've saved it.
Following the second goal, England legend Stuart Pearce was appalled by Liverpool's defending.
He told talkSPORT: "They've only got themselves to blame there. Really high line from Liverpool and I don't know what they're thinking about."
On a night where both historic sides were fielding makeshift back lines, Madrid thoroughly deserved the win.
Without first-choice defenders Sergio Ramos and Raphael Varane, the Spanish giants were only really troubled when Diogo Jota's effort fell at the feet of Salah, who duly converted.
Liverpool's central defensive partnership of Ozan Kabak and Nat Phillips, on the other hand, struggled to cope on Europe's biggest stage and were exposed on numerous occasions.
However, Liverpool have come back from worse and the tie is far from over ahead of the April 14 showdown on Merseyside.
Liverpool were left ruing sloppy defending as Real Madrid seized control of their Champions League quarter-final with a 3-1 victory in the first leg.