Brendan Rodgers' side dominated for the majority of the game but were undone by two brilliant strikes to leave Group A wide open
Anzhi Makhachkala's 2-0 win over Udinese in the early game meant any win for Brendan Rodgers' men would have sealed their progress but they failed to deliver despite a dominant performance on the night.
Joe Cole teed up Jonjo Shelvey to net the opener for the Reds with a close-range header but they were pegged back through a superb strike from Raul Bobadilla after some poor defending.
Luis Suarez was introduced in the hope that he could prevent the Reds needing to win at Udinese on the final matchday and his influence was evident as they took the lead once more following neat link up play with Steven Gerrard and Joe Cole, who slotted home.
It looked like they had secured the precious three points to ensure qualification but Zverotic's rocket two minutes from time deceived an unsighted Pepe Reina and flew into the roof of the net.
Liverpool looked to impose themselves early on and get their passing game going, but without a natural focal point up front, they were stifled early on by the impressive Swiss outfit, who always posed a threat on the counterattack.
Raphael Nuzzolo and Christian Schneuwly both tested Pepe Reina with half-chances as the Bern-based side looked to establish a stranglehold on the match.
Rodgers' side were determined to prove the worth in the increasingly-popular 'false nine' system and clever interchanging saw Shelvey – the furthest midfielder forward - bring the ball down neatly, before releasing Cole, who fired just wide as the home side began to come into the game.
And it was a sumptuous piece of skill from the 'false nine' Shelvey, which almost opened up the resilient Young Boys defence as his flick slipped through the onrushing Jordan Henderson, but his tame effort was straight at Marco Wolfi.
Two substitutions, one for each side due to respective injuries to Dusan Veskovac and Andre Wisdom, threatened to slow the pace of the game down, but Shelvey had other ideas.
The impressive Suso slipped in Cole with a fine pass who, in turn, chipped the keeper to find an unmarked Shelvey, who nodded home from a matter of yards to give Liverpool a lead their first-half performance had largely merited.
It seemed they wanted the win, and with it qualification, wrapped up by the break as they pressed for a second and they came close to doing so when a Suso shot hit the side-netting. From the ensuing corner, Cole's effort hit an oblivious Martin Skrtel, but was blocked on the line by Nuzzolo.
Suso was at the heart of everything good about Liverpool during the half but wasted a gilt-edged chance just moments before the break as he chose to shoot early after another well-worked passing move.
The Reds carried on where they left off as they went close through what was so nearly a carbon copy of their first, as Suso played in Cole, but his chip this time was blocked by the keeper, and within seconds they would pay for their profligacy.
Having narrowly escaped going two down at one end, the Swiss team broke with devastating effect as Alexander Farnerud's cross-field pass found Raul Bobadilla, who escaped the attentions of makeshift right-back Jordan Henderson before lashing across Reina with his left foot on the half-volley.
Rodgers reacted by throwing on talismanic striker Suarez and his decision was almost immediately vindicated as his clever movement allowed Oussama Assaidi to get to the byline, but the final ball was lacking from the lively winger and the visitors eventually cleared their lines.
Suarez's impact would soon tell though. A lovely passage of play between the Uruguayan and fellow sub Gerrard led to the latter feeding in Cole, and the former Chelsea man beat the offside trap to fire underneath Wolfi and regain the lead.
Anfield had fallen quiet in shock after Bobadilla's stunning strike, but having roared back into life after Cole's goal, it appeared they would coast their way to victory.
Young Boys still held hope of gaining the point that would retain their own aspirations of qualifying for the next round and delivered the knockout blow with just minutes left on the clock.
After another swift counter, there appeared little danger as Zverotic took aim from all of 25 yards before unleashing a thunderbolt which flew past the hapless Reina, who may feel he could have done better.
The Reds huffed and puffed for a winner but it proved to be the same old story for them, as they will now reflect on the chances that didn't go in as they prepare for a make-or-break clash in Italy two weeks from now.