Eidur Gudjohnsen put the hosts ahead from the spot after Martin Skrtel brought down Zach Clough in the penalty area, but Raheem Sterling and Philippe Coutinho scored late on to seal victory.
Sterling and Jordan Henderson hit the post, while Emre Can’s rifled effort was pushed onto the bar by an inspired Andrew Lonergan as the visitors tried to draw level with the hosts, who had Neil Danns sent off for a second yellow with 25 minutes remaining.
But Bolton’s last-ditch defending couldn’t hold firm and Sterling and Coutinho ensured Liverpool made the extra man count to set up a last 16 meeting with Crystal Palace – a repeat of the 1990 FA Cup semi-final between the teams that saw current Eagles boss Alan Pardew score the extra-time winner.
It was a low-key opening to the match, but when either side was attacking they both committed men forward, leaving chances to counter at both ends.
Steven Gerrard, playing in his 700th match for Liverpool, had the first effort of the match, drilling a side-footed effort straight at Lonergan before Sterling’s low shot was kept out by the goalkeeper.
The hosts almost took the lead themselves as Wheater rose highest to meet a corner, powering a header down to bring out an excellent stop from Mignolet.
Rodgers’ men kept the ball well and a good move was almost finished off as Lallana played the ball into Sterling, who shifted his body before curling a right-footed shot towards the top corner only to see it hit the post.
Liverpool started the second half well but it was Bolton who took the lead as makeshift strker Matt Mills won Liam Feeney’s cross, and Clough picked up the loose ball before skipping past Skrtel and going down under the defender’s stray foot. Gudjohnsen stepped up and cooly beat Mignolet from 12 yards.
Bolton’s task was made harder when Danns was given his marching orders as – already on a yellow – he flew in on Joe Allen to get a second caution.
It almost didn’t matter as Bolton nearly doubled their lead soon after as Feeney fired in another good cross, but an unmarked Gudjohnsen could only head straight at Mignolet.
With time running out Liverpool hit the frame of the goal for a third time, Can cutting on to his left foot and seeing his curling effort superbly tipped on to the bar by Lonergan, not long after substitute Henderson saw a deflected shot hammer against the upright.
Liverpool’s pressure finally paid as they equalised four minutes from time, Can playing a delightful chip over the top for Sterling to brilliantly fire a volley through Lonergan’s legs.
Rodgers’ men were desperate to avoid extra time, and sealed victory in the first minute of stoppage time when Coutinho cut in from the left and curled a stunning shot in off the crossbar to complete a remarkable comeback and avoid becoming the latest Premier League big gun to crash out of the famous competition.