The Turkish side won the 5-4 on penalties after a gruelling encounter finished 1-1 on aggregate, Croatian defender Lovren seeing his spot kick fly high and wide after nine perfect kicks.
Substitute Tolgay Arslan scored the leveller in the tie in the 72nd minute, curling home left-footed from the edge of the box, before Demba Ba hit the bar in injury time from close range.
Neither side could find a winner, before Lovren miscued from 12 yards to send Besiktas through.
Brendan Rodgers started strikers Mario Balotelli and Daniel Sturridge for only the second time this season, but both struggled to make a mark in the first half.
Balotelli was targeted by Besiktas players throughout, but only retaliated once, earning him a yellow card after just 13 minutes for a late foul on Veli Kavlak.
Liverpool were organised in defence, and the first-half???s best chances fell to Liverpool on the break.
First, Raheem Sterling stung the hands of Cenk Gonen with a half-volley from 20 yards after good work from Alberto Moreno on the left.
Then, after Balotelli played his strike partner clear on the right, Sturridge twisted and turned one too many times, seeing his shot from an angle parried clear of danger.
Gokhan Tore cut inside and hit a decent effort a yard over the bar in the second half, before Arslan hit a ferocious 30-yard effort just inches wide with 20 minutes remaining.
Arslan got his goal with 17 minutes remaining in the game, producing a fine first-time finish from 20 yards after after Ba found him with a flick.
Ba had the chance to win it in injury time, but hit the bar from point-blank range six yards out after the ball dropped to him kindly from a corner.
The former Chelsea forward was then found in extra-time on the right of the area, forcing Mignolet into a decent save before Skrtel cleared near the goal line.
Rickie Lambert, Adam Lallana, Emre Can and Joe Allen all scored in the shoot-out for Liverpool, before Lovren???s woeful penalty sent the Reds packing.