Both sides had good chances to take the lead in the second half; Alex Baptiste forcing a fine save from Simon Mignolet before Kolo Toure hit the post with a header.
Liverpool, who have now lost just twice in 22 games in all competitions, will be looking for their second semi-final appearance in nine years in the replay at Ewood Park.
They were dealt a blow early on as Martin Skrtel was taken off injured having fallen awkwardly after an aerial challenge with Rudy Gestede. Skrtel was knocked out in the incident, but regained consciousness minutes later and was taken to hospital as a precaution.
Liverpool had a penalty shout turned down after 20 minutes as Adam Lallana was played in by Lazar Markovic, but the Englishman was judged to have been thwarted fairly by Matt Kilgallon as he looked to pull the trigger from 10 yards.
Blackburn remained resolute defensively throughout the first half, and had a chance of their own after the half-hour mark.
Gestede???s dummy allowed the ball to roll to an unmarked Craig Conway on the corner of the box, but he lashed his effort wide of Mignolet???s goal.
Liverpool were nearly ahead during a lengthy stoppage-time period for Skrtel???s earlier injury, but Sturridge???s effort from 15 yards having rolled Kilgallon inside the area was punched clear by Simon Eastwood.
Blackburn came out flying in the second period, and were denied by an acrobatic Mignolet save from an Baptiste header.
Substitute Toure also went close with a header at the other end, hitting the post after a fine deep delivery from Philippe Coutinho.
Captain Jordan Henderson forced Eastwood into another save with five minutes remaining, seeing his volleyed effort from an angle cleared by the feet of the Rovers goalkeeper.
Sturridge and substitute Mario Balotelli missed late half-chances for the seven-time Cup winners, but Blackburn held on for the replay for a chance of a Wembley semi-final appearance.