Brendan Rodgers' men suffered a chastening 3-1 loss in south-east London earlier this season and the hosts suggested a repeat when Fraizer Campbell gave them a 15th-minute lead.
But Liverpool had only lost twice in 21 games across all competitions since that reverse last November and they brushed off a frustrating first half when Daniel Sturridge equalised with a fine volley after 49 minutes.
Dwight Gayle came off the bench to score twice in the dramatic 3-3 draw that dealt a decisive blow to Liverpool's Premier League title bid last season and the recalled forward was instrumental in the game's opening goal.
Speroni was called upon again with half an hour played as Jordan Henderson arrowed in a swerving drive a minute before Palace debutant Pape Souare's clumsy challenge on Sturridge in the area went unpunished.
The patient build-up play Liverpool displayed throughout the first half was rewarded early in the second when Henderson floated a ball to the back post for Sturridge to volley firmly through Speroni's legs.
Palace bustled forward looking to restore their lead and Mignolet reacted well to turn Gayle's bobbing strike behind after a corner was partially cleared, but the turnaround was completed in the 58th minute with Balotelli to the fore.
Some quick footwork allowed the Italy international to draw a foul on the edge of the box from Scott Dann and Speroni could only block his resulting low free-kick into the path of Lallana, who made no mistake.
Set-pieces appeared to be Palace's best route back into the match as Mignolet punched uncertainly in the 63rd minute.
But it was a rare blot on an assured performance from the often-maligned Belgian goalkeeper, who was booked for time-wasting in stoppage time but played a full part in seeing Rodgers' team through to the last eight.
