The Championship team, who fielded the veteran strike pairing of Eidur Gudjohnsen and ex-Liverpool man Emile Heskey, proved tireless opponents and made it a frustrating evening for their hosts.
Javier Manquillo passed up the clearest opening for Liverpool, while a lively Raheem Sterling was regularly thwarted by last-ditch challenges from a disciplined Bolton defence - behind which goalkeeper Adam Bogdan excelled.
As he contends with League Cup and Europa League campaigns alongside Premier League commitments, the return game at the Macron Stadium is unwelcome for Liverpool manager Brendan Rodgers, but a preferable fate to the shock defeats endured by fellow heavyweights Manchester City and Chelsea on a memorable day of FA Cup action.
Bolton made a confident start, with Matt Mills seeing an 11th-minute strike blocked before the defender returned to his own penalty area to deny Adam Lallana - making a first Liverpool start since New Year’s Day - with a perfectly timed challenge.
Bogdan extended himself to keep out a sweetly struck 30-yard effort from Jordan Henderson, while Mills was on hand to frustrate Sterling after a lofted ball forward exploited a high Bolton defensive line.
In the 33rd minute, Dean Moxey was inches away from putting the Championship team ahead - his half-volley from the edge of the penalty area whistling narrowly past Simon Mignolet's right-hand upright.
Like Mills, Tim Ream was enjoying an impressive outing at centre-back for Bolton and the USA international stopped Lallana from bundling home.
Glen Johnson and Lallana then fired rasping strikes into Bogdan's arms as Liverpool began to find their rhythm during the closing stages of the first half.
Rodgers sent on Lazar Markovic in place of Jose Enrique at half-time and Sterling tore towards the Bolton area within a minute of the restart, but Mills was on hand to block the shot.
Mills was nursing a yellow card for an earlier foul on Sterling and there was an anxious moment for the former Reading man when Markovic went to ground under his challenge. Referee Kevin Friend was unmoved, prompting a furious reaction from Rodgers on the touchline.
Bogdan was out bravely to save at Sterling's feet when Philippe Coutinho unpicked the Bolton defence with a 53rd-minute throughball and the goalkeeper made another fine stop by getting down to push a Coutinho drive away from the bottom corner.
Moxey stumbled as he looked to deal with the rebound but Liverpool wing-back Manquillo pulled a glorious chance wide.
Youngster Conor Wilkinson was introduced in place of Heskey but Gudjohnsen remained and spurned a pair of chances in quick succession.
The former Chelsea and Barcelona man side-footed wide from Ream's low cross before more wastefully blazing over on the end of Wilkinson’s 64th-minute cutback.
Manquillo made way for Fabio Borini as Liverpool chased victory and the Italian forward forced Bogdan into another splendid stop with 19 minutes to play.
Borini headed agonisingly wide in stoppage time and Lucas Leiva saw Bogdan beat away a fierce drive as Neil Lennon's team claimed the replay they craved.
