The Reds opened the scoring in the fourth minute through a fine solo effort from Raheem Sterling.
Sturridge then made the most of a defence-splitting pass from Steven Gerrard to put the hosts 2-0 up in the 29th.
That lead was the very least Liverpool deserved from a first half they utterly dominated and which also saw Gerrard and Daniel Agger hit the woodwork.
Their goalkeeper Simon Mignolet had been called upon shortly after Sterling's effort, though, to keep out a Joss Labadie free-kick.
And Sky Bet League One outfit County then went at their opponents early in the second half and were rewarded with a headed goal in the 62nd minute from Yoann Arquin.
Liverpool looked to have regained control of the contest from there and Kolo Toure became their third player to hit the post.
But the visitors then stunned Reds fans by snatching an equaliser with six minutes of normal time remaining, substitute Adam Coombes slotting in.
County had penalty shouts turned down in extra-time when Enda Stevens went down under the attentions of Sterling in the box, and moments later Sturridge stroked in his second goal of the night and fourth of the season.
That strike was in the 105th minute and in the 110th, substitute Jordan Henderson cut through the County defence to wrap up the win.
County were under pressure from the word go and Liverpool soon had the ball in the net.
Sterling, having collected the ball from Glen Johnson midway into the Magpies' half, proceeded to skip through their defence and into the box before firing home past goalkeeper Bartosz Bialkowski.
Three minutes later Gerrard sent a free-kick curling off-target.
But it was the visitors who then might have scored through a free-kick of their own, drilled by Labadie to bring a decent save out of Mignolet.
Aly Cissokho, on his first Liverpool start, had to come off soon after with an injury and was replaced by Agger.
Normal service then resumed as Gerrard powered over the bar from distance and then whipped a free-kick into the area which Agger headed against the post.
Joe Allen teed up Sturridge, who lifted a shot just beyond the corner of the net.
And after Sterling had also shot too high, the second goal duly arrived just before the half-hour mark as Sturridge collected Gerrard's delightful through-ball and cracked a shot in.
Agger's header from a corner was well kept out by Bialkowski, and Gerrard then struck the post having been put in on goal by Sterling.
County emerged after half-time looking intent on making an impression and after Labadie had sent a weak header wide, Callum McGregor drove into Mignolet's arms.
Bialkowski had little trouble dealing with a tame Glen Johnson effort and County were then suddenly back in the contest.
Manny Smith flicked a free-kick to Arquin and the striker beat his marker Andre Wisdom to nod past Mignolet.
That appeared to shock Liverpool into life and in the 69th minute the ball came to Toure, but he could only shoot against the post.
Sturridge then headed wide but County were not done, and in the 84th minute Coombes found space in the box to finish from Arquin's cut-back and send the away section of Anfield into raptures.
Liverpool desperately scrambled for a late winner, with Agger's shot deflecting wide and Sturridge trying a clever chip that did not work.
The game went into extra time and the Reds, having made all three substitutions, were reduced to 10 men when Toure was carried off on a stretcher.
Towards the end of the first period Stevens went to ground in the area after contact from Sterling, but no penalty was awarded.
And soon after, Sturridge converted into the far corner, with Henderson then surging through County and applying a cool finish to restore the two-goal advantage.