Hulk had given the visitors a 19th-minute lead after capitalising on a disastrous error of judgement from Jamie Carragher, before the Uruguayan and Joe Allen struck to give Liverpool hope at half-time but could not push on from Suarez's second, a brilliant free kick, to progress.
The visitors, meanwhile, installed Sergey Semak in favour of top-scorer Aleksandr Kerzhakov, with Hulk moving into a central attacking role.
The Kop was in full voice and boos greeted every Zenit touch, while Liverpool showed the same poise and control in possession that helped them thrash Swansea City 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 hosts a glimmer of hope minutes later, thumping home a low free kick from the edge of the area.
And Liverpool were not done there as a fantastic cross from Jose Enrique found Allen on 43 minutes, who, after having his initial shot saved, reacted brilliantly to fire home the rebound from inside the six-yard box. Anfield was rocking once again.
The visitors made two changes at the break but it was all Liverpool from the restart and their incessant pressure suggested the required target of two more goals was well within their reach.
|8/1||Liverpool are 8/1 with William Hill to finish in the Premier League top four
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.
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.
And the goalkeeper 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 the Reds could provide no fairytale as Zenit held on to ensure that Liverpool's participation in Europe this season came to an admirable end.