The Reds won 3-1 on the night thanks to a tremendous performance at Anfield, and while that left the score 3-3 on aggregate following a 2-0 loss in Russia last week, it was Zenit who progressed after Hulk collected an early away goal.
Joe Allen was the other scorer on a night that came close to emulating famous comebacks such as Olympiacos 2004 and Auxerre 1991.
Liverpool started brightly amid a wall of noise at Anfield, Steven Gerrard and Jordan Henderson mustering early shots.
It was a night when an early goal would be required — but unfortunately it came at the wrong end, Hulk capitalizing on a mis-hit back pass from Jamie Carragher and slotting past Pepe Reina at the Kop end.
Nineteen minutes gone and Liverpool now required four goals to progress, but Anfield did not lose hope.
In the 28th minute, the arrears were reduced after Daniel Agger slalomed through the visiting defense and drew a foul inches outside the area. Suarez stepped up, navigated both wall and goalkeeper and caused an eruption in L4.
Moments later Gerrard had another dead ball cleared off the line by Aleksandr Anyukov.
The comeback continued in the 43rd minute when a superb combination between Henderson and Jose Enrique granted the latter space to cross. The Spaniard found Allen, who lifted the ball over the goalkeeper after having an initial header saved.
The second half began with a penalty appeal after a handball by Tomas Hubocan. Referee Bjorn Kuipers waved play on and Suarez found the ball at his feet only to fire wide as the crowd continued to protest.
Never mind. The third goal arrived after 59 minutes as Suarez bent home his second free-kick of the night, this time from 30 yards at the Kop end. What a player, and what a night this was turning into.
Both Oussama Assaidi and Jonjo Shelvey had been preparing to come on when the goal went in and Brendan Rodgers stuck to his plan, replacing Henderson and Allen.
The belief inside Anfield was palpable as the visiting defense crumbled in the face of a Red onslaught. Hulk pumped a free-kick of his own towards Reina’s goal but the Russians were otherwise impotent in attack.
Another free-kick for Liverpool, this time after Assaidi was hauled to the ground. Suarez stepped up for a hat-trick but this time was denied by Vyacheslav Malafeev. Seconds later the goalkeeper made an acrobatic stop to thwart a Gerrard volley.
At the other end Hulk forced Reina into a rare save but Liverpool soon surged again, Shelvey blasting into the side netting. Agger then skimmed a header wide during a succession of corners. Time was beginning to run out.
Despite continued pressure, it was not to be for Liverpool, whose elimination means that Carragher’s 150th European appearance will be his last.