The visitors had a number of decent opportunities in a first half, which saw the Reds losing the services of Martin Skrtel early on after he landed awkwardly following an aerial challenge with Rudy Gestede.
Brendan Rodgers's unit piled on the pressure after the break but the likes of Daniel Sturridge and Raheem Sterling were unable to find the target at Anfield.
Manchester United on Monday will host Arsenal in the last-eight of the competition, while Reading and Bradford will contest a replay on March 16 after drawing 0-0 at Valley Parade on the weekend.
Aston Villa are the only side through to the semi-finals following a 2-0 win over West Bromwich Albion on Saturday.
Fabian Delph and Scott Sinclair got the goals at Villa Park as both teams ended the match with ten men.
The Baggies' Claudio Yacob was sent off for a second yellow card in the 80th minute, while Villa's Jack Grealish was dismissed at the death after being cautioned twice in three minutes.