The holders looked to have ended Atletico's dogged resistence when Roberto Firmino put them ahead in the tie after Georginio Wijnaldum's first-half header forced the game into an extra 30 minutes.
However, Liverpool were made to rue a host of missed chances to kill the tie off in 90 minutes as Llorente twice curled home from outside the box before Morata twisted the knife by breaking away to score in added time of extra time.
However, it remains to be seen if and when the last eight will even be able to take place due to the chaos caused across the continent by the new coronavirus.
A packed Anfield played its part in trying to roar Liverpool onto another famous Champions League conquest as, unlike in numerous countries across Europe, no restrictions have so far been put on supporters attending games in the United Kingdom.