It took Sturridge just seven minutes to open his account for the Reds. Jonjo Shelvey played in the England international, who used his pace to get beyond Exodus Geohaghon, before slotting the ball beyond Alan Marriott in the home goal.
The £12m signing from Chelsea could have had a second with 18 minutes played when he found himself with only Marriott to beat, but the Mansfield goalkeeper got down to flick the ball away from the attacker.
© AP Photos
Liverpool continued to dominate the tie, but full-back Andre Wisdom was denied by a last-ditch block, before Marriott proved to be equal to a rising shot from Shelvey.
In what proved to be a rare venture forward, Mansfield almost levelled up proceedings with 35 minutes on the clock. Geohaghon picked out Matt Green on the edge of the area and the attacker then saw his powerful effort tipped to safety by Liverpool's stand-in keeper Brad Jones.
The remainder of the half belonged to Liverpool, but they were unable to add to their tally.