Moses settled some Liverpool jitters at a packed Dean Court, scene of Bournemouth's famous FA Cup defeat of Manchester United in 1984, with a low finish after 25 minutes.
Sturridge then combined with Luis Suarez to finish off the hosts after an hour.
However, the visitors were matched by Bournemouth throughout the match and survived a few scares before halftime.
Lewis Grabban started and finished a flowing Bournemouth move only to see his shot deflected over the crossbar by Kolo Toure and shortly afterwards Tommy Elphick headed a great chance over the bar from a corner.
Bournemouth fell behind shortly afterwards when Moses, on loan from Chelsea, received the ball just outside the penalty area, took a step inside and drilled a right-foot shot through the legs of a defender and just inside the post.
It was tough on Bournemouth who kept plugging away and were still very much alive in the early stages of the second half when they were denied a penalty after Liverpool defender Martin Kelly blatantly grabbed the shirt of Simon Francis.
Their sense of injustice was heightened almost immediately when Suarez slipped a perfectly weighted pass through the Bournemouth rearguard and Sturridge ran on to score.
Liverpool join Arsenal in the draw for the fifth round after the Gunners beat Coventry City on Friday.
Later on Saturday several more Premier League sides are in action with Manchester City at home to Championship side Watford and Everton away at Stevenage.
Chelsea host Stoke City on Sunday.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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