Steven Gerrard's 700th Liverpool appearance looked set to end in defeat when a 59th-minute penalty from Eidur Gudjohnsen put hosts Bolton ahead in Wednesday's fourth-round replay.
However, Raheem Sterling's fine volley pulled Liverpool level with four minutes remaining - following the dismissal of Neil Danns - and the Championship side were subjected to further heartbreak when Coutinho, who signed a new long-term deal at Anfield earlier this week, found the net with a sublime 25-yard strike.
Defeat would have been harsh on Brendan Rodgers' men, who hit the woodwork on three occasions prior to their late goals.
Liverpool will now travel to Crystal Palace in round five, while Bolton will return to Championship action having given their more illustrious opponents a major scare over 180 minutes.
Home goalkeeper Andy Lonergan had to be alert early on to deny Gerrard - who followed Ian Callaghan and Jamie Carragher in reaching 700 games for Liverpool - and Sterling as the visitors made a fast start.
Bolton gradually grew into the game, and Gudjohnsen should have done better when he was played in by Josh Vela, only to shank his effort high over the bar.
The underdogs could well have taken the lead when David Wheater connected with Liam Feeney's free-kick, but the centre-back's header was well held by Simon Mignolet.
Liverpool responded by going even closer to an opener, as Sterling picked up the ball outside the area and curled a superb effort that rattled off the crossbar.
Mignolet was called into action again to keep out a dipping half-volley from Bolton's joint top-scorer Matt Mills, who led the line rather than occupying his usual position in defence.
Liverpool threatened to seize control as the opening half drew to a close though, with Coutinho – looking bright cutting in from the left flank – driving just wide of the right-hand upright from the edge of the box.
Bolton made a strong start to the second period and were buoyed further when Sterling started limping, with an apparent recurrence of an ankle knock picked up in the first half.
That didn't stop the forward twice testing the Bolton defence – breaking through only to be halted by Dorian Dervite's terrific last-ditch tackle before forcing Lonergan into another fine stop.
Liverpool were left stunned when referee Roger East penalised Martin Skrtel for a foul on Zach Clough in the area, with Gudjohnsen coolly side-footing home the resulting spot-kick.
However, Bolton then suffered a setback when Danns picked up a second yellow card for a late challenge on Joe Allen and Liverpool twice hit the woodwork, through Jordan Henderson and Emre Can, as they surged forward in search of an equaliser.
Former Bolton forward Daniel Sturridge was introduced by Rodgers, but it was Sterling who eventually equalised with a controlled left-footed volley from Can's exquisite pass.
Coutinho then took centre stage, firing a dipping shot beyond Lonergan to leave Bolton devastated.
