Jurgen Klopp's defending European champions trailed 1-0 from last month's first leg in Madrid but levelled up the tie through Georginio Wijnaldum's header two minutes before half-time in Wednesday's dramatic match.
Inspired Atletico goalkeeper Jan Oblak made a string of saves either side of Wijnaldum's breakthrough goal and Andy Robertson headed against the crossbar as the contest went the distance at Anfield.
Roberto Firmino put Liverpool ahead in the tie for the first time with a goal early in extra time, but Llorente - brought on in the second half - capitalised on an Adrian error to swing the game back in Atleti's favour and added a second eight minutes later.