The Uruguayan starred with two sensational free kicks either side of a Joe Allen strike, but the home side ultimately fell short of another memorable European comeback.
Hulk had given the Russian side a 19th minute lead after capitalizing on a disastrous error in judgement from Jamie Carragher, before Suarez and Joe Allen struck to give Liverpool a halftime lead.
Suarez scored again after the break but the Reds ultimately fell just short of qualification as they were left to rue Carragher’s first-half mistake, as well as their first-leg performance.
Liverpool was without cup-tied duo Daniel Sturridge and Phillipe Coutinho, while Daniel Agger replaced the injured Martin Skrtel alongside Carragher, who was making his 150th European appearance.
Zenit, meanwhile, installed Sergey Semak in favor of top-scorer Aleksandr Kerzhakov, with Hulk moving into a central attacking role.
After the first leg, Brendan Rodgers had given his side a “fighting chance” of progression and the Kop was in full voice in the buildup to kickoff.
Boos greeted every Zenit touch, while Liverpool showed the same poise and control in possession that helped it thrash Swansea 5-0 on Sunday.
But on 19 minutes, disaster struck. Carragher was caught in possession inside his own half by Hulk, who suddenly had 40 yards of space to himself with the Liverpool goal gaping. The Brazilian burst forward and calmly slotted past Pepe Reina – scoring his sixth goal for Zenit and second of the tie.
A steep hill had suddenly become a mountain as Carragher’s catastrophic error meant Liverpool now needed four goals to progress.
But Suarez offered the home side a glimmer of hope minutes later, thumping home a low free kick from the edge of the area.
Liverpool wasn’t done there as a fantastic cross from Jose Enrique found Allen on 43 minutes. After having his initial shot saved, the midfielder reacted brilliantly to fire home the rebound from inside the six-yard box. Anfield was rocking once again.
The Russians made two changes at the break, but it was all Liverpool from the restart and the incessant pressure suggested the required target of two more goals was well within reach.
Zenit was feeling the heat. After 50 minutes, Liverpool was refused a penalty even though Tomas Hubocan visibly handled in the area, before Suarez was then denied by the palms of Vyacheslav Malafeev.
Suarez could not be stopped on 58 minutes, however, as the Uruguayan curled a sensational 25-yard free kick into the top corner for his second – and Liverpool’s third – of the night.
Unbelievably, the 26-year-old almost made it a hat trick of free kicks 10 minutes later, but Malafeev was this time able to punch the ball away.
Malafeev was Zenit’s hero again moments later as he dived to save a Gerrard half volley, before Reina kept Liverpool in the tie with a save of his own straight up the other end.
The clock was ticking, though, and the tension mounting. The stage was set for a memorable finale.
But there would be no fairytale ending for the Reds as Zenit held on to ensure Liverpool’s participation in Europe this season had come to an admirable end.