After an uneventful first half, Bolton’s 36-year-old striker Eidur Gudjohnsen had two big opportunities to win the game in the second, hitting wide and then over from 15 yards.
Goalkeeper Adam Bogdan was Bolton’s hero on the night, keeping out Philippe Coutinho’s low drive and then Fabio Borini’s effort from inside the area in the latter stages.
Liverpool will now have to travel to Greater Manchester for a fourth-round replay next month.
Neil Lennon started with Gudjohnsen and his 37-year-old striker partner Emile Heskey, returning to the club he once enjoyed a fruitful spell with between 2000 and 2004.
Chances were few and far between in the first half, but Raheem Sterling nearly gave the hosts a lead after half an hour, seeing his effort from inside the area blocked well by Matt Mills.
A few minutes later, Bolton went up the other end through Dean Moxey, who fired just wide on the half volley after the ball fell to him 25 yards out.
The visitors should have been reduced to 10 men in the second period, but referee Kevin Friend decided not to penalise Mills, already on a booking, despite bringing down Lazar Markovic just outside the area.
Liverpool had two quick-fire chances just before the hour mark; first Coutinho’s curling effort from 20 yards was parried away by Bogdan, then Javier Manquillo got ahead of Tim Ream to volley just wide from eight yards when he should have hit the target.
Gudjohnsen then missed the first of his two chances from 15 yards after being found by Ream, side-footing wide of Simon Mignolet’s goal, before firing over a volley from Dorian Dervite’s centre with 23 minutes remaining.
Reds substitute Fabio Borini was found clear inside the area by Coutinho on the counter attack, but his placed effort was superbly turned around the post by Bogdan, before Adam Lallana blazed over the bar after good work from Sterling and Coutinho.
And Liverpool nearly grabbed the winner at the death, but Borini’s header went narrowly wide of Bogdan’s goal in injury time as Lennon’s side held on for a deserved replay.