The Reds looked on course for a comfortable passage into round three of the competition as they surged into a two-goal lead courtesy of first-half efforts from Raheem Sterling and Sturridge.
But the Magpies fought back when Yoann Arquin nodded home from close range just past the hour.
And they sent their travelling support into ecstasy six minutes from time when substitute Adam Coombes fired in the leveller with his first touch.
Liverpool were then forced to play the majority of extra-time with 10-men when Kolo Toure was stretchered off and they had made all of their allotted changes.
But Brendan Rodgers' side proved too strong for Chris Kiwomya's men as Sturridge and Jordan Henderson wrapped up a hard-fought triumph.
However, the victory came at a cost for Liverpool as new signing Aly Cissokho, Joe Allen and Toure all went off injured ahead of Sunday's crunch clash with Manchester United.
Having suffered early exits at the hands of Northampton Town and Swansea City in recent years, Rodgers opted to name a strong starting XI for the visit of the League One strugglers.
There was a clear sign of intent from the Merseysiders and they raced out of the traps, grabbing the lead on four minutes.
Sterling found space just outside the area and used the clever run of Sturridge to outfox two defenders before drilling a low shot into the corner via the hand of Bartosz Bialkowski.
The visitors offered a brief riposte when Joss Labadie tested Mignolet with a thumping free-kick at the Kop end, but from then on it was one-way traffic.
Daniel Agger saw a glancing header crash back off the inside of the post following an inswinging free-kick from Gerrard while youngsters Sterling and Jordan Ibe impressed on the wings.
The Magpies had no answer to wave after wave of attacks from the hosts and on the half hour mark it was two.
Gerrard picked up possession deep in County territory and slid an exquisite pass into Sturridge on the left side of the area. The in-form striker took one touch before despatching a fierce drive into the far corner.
The Reds continued to pile forward and Agger forced a stunning save out of Bialkowski while Gerrard cracked the near post with a shot from a tight angle.
County could have feared what lay ahead in the second period, but they fought back valiantly and restricted Liverpool to fewer opportunities as the half progressed.
Rodgers would no doubt have been considering withdrawing the likes of Gerrard and Sturridge but he was forced to shelve those plans on 62 minutes when the League One side reduced the arrears.
A free-kick was pumped high into the home team's area where Manny Smith rose brilliantly to nod it back across goal for Arquin to head beyond Mignolet.
Liverpool responded and almost made it three when Toure side-footed a Gerrard cross against the post from inside the six-yard box.
It would prove a costly miss as the away side sprung forward and snared an unlikely leveller with just six minutes left on the clock.
Arquin broke the offside trap down the right and rampaged into the area. The striker drove to the byline and fizzed the ball across for Coombes who enjoyed the finest first touch of his career by coolly steering it into the net.
County fans were beginning to dream of an upset and in extra-time Liverpool suffered a further blow when Toure was stretchered off, leaving them to finish the match with 10-men.
The League One side were suddenly in the ascendancy and the Reds survived a big penalty shout on 104 minutes when Enda Stevens tumbled in the area under pressure from Sterling.
The referee waved away the appeals and the hosts broke forward to snatch the lead once again courtesy of a thumping drive from Sturridge after some great approach play by Coutinho.
And they made sure of progress in the second half of extra-time when Henderson seized upon a pass in the centre circle, slipped the ball through the legs of Gary Liddle and then raced through to fire a low shot into the net.
MAN OF THE MATCH
Daniel Sturridge (Liverpool): The striker continued his rich-vein of form with a well-taken double and got his team out of trouble with the all-important third goal of the night.
LIVERPOOL: Mignolet 6, Johnson 6, Toure 6, Wisdom 6, Cissokho 5, Allen 6, Gerrard 7, Alberto 6, Sterling 7, Ibe 7, Sturridge 8. Subs: Agger 7, Henderson 7, Coutinho 7.
NOTTS COUNTY: Bialkowski 7, Dumbuya 7, Stevens 7, Liddle 7, Smith 7, McGregor 8, Bell 6, Labadie 7, Campbell-Ryce 6, Fotheringham 6, Arquin 8. Subs: Thompson 6, Boucaud 6, Coombes 7.
- Sports & Recreation
- Daniel Sturridge
- League One
- Steven Gerrard
- Notts County
- Kolo Toure
- Simon Mignolet