Atletico had won just one of its last seven matches ahead of this tie, but it took the game to Liverpool from the off and opened the scoring through Saul's fourth-minute strike.
Returning to the scene of its magical sixth European Cup triumph in June, the visitor dominated possession but failed to register a single attempt on target in response.
Just a third defeat in all competitions this season leaves Jurgen Klopp's men with it all to do in the return fixture at Anfield on 12 March.
Koke's set-piece delivery bounced off Fabinho and squirmed through for Saul, who stabbed the ball past Alisson from point-blank range to give Atletico an early lead.
Atleti pushed for a second goal and an error from Virgil van Dijk gifted Alvaro Morata a free shot from a tight angle, but Alisson was equal to the low attempt.
Liverpool took time to grow into the contest and had the ball in the net through Mohamed Salah after 26 minutes, only for Roberto Firmino to be flagged for offside in the build-up.
Salah flashed a header wide from Liverpool's next big chance eight minutes into the second half as the host continued to withstand the pressure.
Jordan Henderson guided a first-time effort just wide before limping off injured in the final 10 minutes as the reigning champion was left frustrated on its return to Madrid.