Raheem Sterling opened the scoring after four minutes with a fine solo effort before Sturridge grabbed his first of the game with a powerful drive into the far corner.
Notts County pulled off an unexpected second-half comeback to force extra time, however, with Yoann Arquin heading the first in off the crossbar before setting up Adam Coombes to calmly stroke the equaliser in from close range.
Liverpool were reduced to 10 men in extra time when, having made all three subs, Kolo Toure was stretchered off, but Sturridge restored his side's lead with a breakaway goal from a tight angle.
Jordan Henderson then secured Liverpool's place in the third round with a neat finish into the bottom corner after a powerful run.
Here, Sports Mole takes an in-depth look at an action-packed 120 minutes of football at Anfield.
On target 9
On target 5
Was the result fair?
Yes. Liverpool were the better team for the vast majority of the match and they really should have won inside normal time. At half time they were cruising and there looked to be no way back for Notts County, but a combination of luck and ruthlessness in front of goal saw them pull off an unlikely comeback. The visitors deserve credit for their performance, and their spirit will have earned them plenty of admirers, but there is no doubt that Liverpool deserved to win the match.
Manager Brendan Rodgers will come away from this match with mixed feelings. On the one hand, his side are safely through to the next round, looked very good in the first half and coped well with being reduced to 10 men. On the other hand, they suffered three injuries ahead of Sunday's match with Manchester United, failed to close out what should have been a routine win and conceded two sloppy goals to a League One side. For large parts of the match, particularly in the first half, Liverpool's performance was very assured, but they took their foot off the pedal in the second half and their failure to kill the game off inspired Notts County. Plenty of positives, but also plenty of room for improvement.
Notts County's performance
The fact that they will be gutted not to have won this match tells you something about their performance. They were outclassed in the first half but simply didn't give up in the second. They didn't have many chances, but they were clinical in front of goal when they needed to be. They did ride their luck at times, with Liverpool hitting the post on three occasions, but to score two goals against Liverpool at Anfield requires more than just luck. They dominated possession when Liverpool were reduced to 10 men and became more of an attacking threat, but in the end Liverpool's superior quality shone through. Even so, on that performance it is hard to see why they are in the League One relegation zone.
Sports Mole's man of the match
Daniel Sturridge: It is tough to choose a man of the match in such a topsy-turvy game, but Sturridge was the one who ultimately made the difference. The England international has been in scintillating form so far this season and he took his two goals very well. He also created the space for Sterling to run in to for his goal, so can feel like he has an assist of some sort too. His second goal was particularly important as Liverpool were coming under increasing pressure in extra time and needed something to stem the tide.
This one goes to the Liverpool defence for their part in Arquin's goal. The Notts County striker was the only one who looked like he really wanted to win the ball on the edge of the six yard box as Toure and Agger stood watching it loop in the air. It was not commanding play from either defender, and Simon Mignolet could have perhaps come off his line too. Poor play all round from Liverpool, and it gifted Notts County their route back into the match.
The referee was very rarely noticed tonight, and that is always a good thing. He had two big penalty calls, both of which were fairly similar and both of which could have gone either way. The worst thing he could have done is give two different decisions, but he stuck to his guns and called no penalty on both, despite the latter coming for Notts County in injury time when it could have been easy to get caught up in the moment.
Liverpool: Up next for Liverpool is a Premier League match against their arch rivals Manchester United on Sunday as they look to maintain their 100% league record.
Notts County: At the other end of the scale, Notts County will be looking for their first league win of the season when they host Rotherham United on Saturday.