Now Diego Simeone's team must compete with Porto and AC Milan for second place in the table.
Trent Alexander-perfectly Arnold's timed cross from far on the right broke open a five-man defence and was nodded in by Jota after just 13 minutes of play.
Mane scored a sidefooted finish to a move that the defender started with a superb drive into the middle of the field.
Atletico Madrid's dreams of recovering a two-goal deficit in the same fashion as they did in Madrid were dealt a significant blow after the half-hour mark.
A straight red for Felipe instead of a booking was the result of him refusing to walk to the referee after he deliberately brought down a breaking Mane following a corner.
In the second half, a Jota goal was ruled out for offside before Mohamed Salah and Brendan Rodgers both wasted good chances. Afterward, Liverpool's numerical advantage was evident.
There looked to be little hope for hosts when Luis Suarez's deflected shot was saved by VAR after Jose Giminez had taken up an offside position before passing the ball to his Uruguayan teammate, who then diverted the ball into his own net.
With 15 minutes remaining, Thiago Alcantara's careless play set up Hector Herrera for a shot that sailed over the crossbar. And he missed.
There were no further alarms for Liverpool, except from the injury to replacement Roberto Firmino, as they completed an outstanding victory over LaLiga champions Real Madrid
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