Having been reduced to ten men following two yellow cards for Lucas Leiva, Liverpool were eventually beaten when Dan Gosling scored in the 118th minute.
Having only celebrated his 19th birthday two days ago the teenage hero bagged his second goal for the Blues and set up a home tie with Aston Villa in the next round.
A bright start from the home side had Everton chasing down and pressuring the ball deep inside the Liverpool half but the visitors didn't take long to settle and a half-volley from Xabi Alonso just outside the box flew just wide of a relieved Tim Howard.
Just when the Reds looked to be getting a bit of fluidity about their game, activity from the bench indicated an early substitution and captain Steven Gerrard was replaced by Yossi Benayoun after only 20 minutes due to a pulled hamstring.
The Blues instantly took hope from the huge missing piece of the Liverpool side and started stringing a few passes together in search of an opening through the compact final third of the pitch.
Crosses from both wings tested the heading abilities of Martin Skrtel and Jamie Carragher in the centre of the visiting back four but even with a well-struck Leon Osman volley inside the box after half-an-hour, Pepe Reina had still to make a save.
Struggling to find any continuity in what was turning out to be a scrappy battle, Liverpool looked unsettled by the physical side of the game and they certainly made their feelings heard as referee Alan Wiley left the field at half-time with neither side looking a goal threat.
With no substitutions by either manager at the break, the second half began much in the same way as it left off with the game confined to the packed middle third of the park.
The first chance fell to Rafa Benitez's men when a defence-splitting pass from Xabi Alonso caught the Everton back line napping and Albert Riera found himself unmarked inside the box but, having waited for the ball to roll across his body, Howard was able to sprint off his line and block the close-range shot.
In the 70th minute, after a quick Everton counter-attack, Mikel Arteta found Tim Cahill on the penalty spot and his neat first-time flick to Osman placed the ball on the midfielder's favoured right boot but the shot bounced back off Reina's near post and away from danger.
A huge turning point came when Liverpool were reduced to ten men with quarter-of-an-hour of normal time still remaining when Lucas Leiva brought down Joleon Lescott with a late challenge.
Everton started to use the extra-man advantage, forcing Liverpool to chase the ball but, having not tested Reina all night, David Moyes' men couldn't find a pass to open up the solid Liverpool back-line and the 90 minutes ended with the sides still locked at 0-0.
Moyes' men flew out of the starting blocks at the start of extra-time and could have scored when a left-footed cross from Leighton Baines curled away from the keeper but Cahill couldn't stretch his leg far enough around Skrtel to get the telling touch.
Everton were throwing wave after wave of attacks at Liverpool and Reina produced an excellent double save when he pushed Arteta's drive and Gosling's follow-up out for a corner which was met by Cahill only for his header to bounce inches wide.
Everton almost broke the deadlock when young Gosling drew the foul out of Alonso with a clever back-heel and, after placing the ball, Arteta struck it low and hard with his shot almost ending up in the bottom corner.
With two minutes left on the clock both sides looked destined for penalties but a deep cross from Andy Van der Meyde fell to 19-year-old Gosling who got the ball on to his right boot and struck it against Alvaro Arbeloa and over a stranded Reina.