Daniel Sturridge and Jordan Henderson saved the day after a second-half revival from the visitors, but the withdrawals of Aly Cissokho, Kolo Toure and Joe Allen worried the Reds.
In-form striker Sturridge netted his second of the game in the 105th minute before his midfield teammate sealed the Premier League outfit's progress to the third round five minutes later with a cool finish.
Those goals came after second-half strikes from Yoann Arquin and Adam Coombes had brought the spirited League One visitors level, following Raheem Sterling's early opener and Sturridge's first goal before the break.
|By David Lynch at Anfield
"Liverpool fans would have expected an easy ride in the second half, with their side cruising at 2-0 up, but the admirable Notts County comeback brought them a rather unwanted extra 30 minutes ahead of Sunday's meeting with Manchester United.
"They may have eventually pulled through to seal their place in the next round but injuries to Kolo Toure, Joe Allen and Aly Cissokho represent the true cost of Tuesday's action."
Liverpool unsurprisingly dominated the early stages and were ahead after just four minutes through Sterling. The 18-year-old winger collected a pass from Johnson 30 yards from goal and jinked past two defenders before firing low past Bartosz Bialkowski.
Rodgers's men continued to control proceedings and substitute Daniel Agger - who had replaced the injured Aly Cissokho early on - hit the inside of the post after glancing Gerrard's set-piece delivery goalwards on 19 minutes.
Sturridge's movement was causing all sorts of problems for the Notts County back line and the 23-year-old was rewarded with his third goal of the season and his ninth in his last eight appearances after 29 minutes with a clinical finish. Gerrard was again the architect as he guided a perfect pass to the striker, who took one touch before firing past the helpless Bialkowski into the top corner.
The Liverpool skipper himself then almost added a third four minutes before the interval, lashing a left-footed shot off the post after Sterling had found him with a well-weighted pass.
Liverpool then hit the woodwork for a third time as Toure met another Gerrard delivery, but he could only strike the post when it looked easier to score. The home side was made to pay for its profligacy when Coombes equalized - netting his first senior goal - after good work from Arquin to force extra time.
Notts County began the additional 30 minutes the brighter of the two sides and its chances were boosted further when Toure was carried off with a groin injury, leaving Liverpool with 10 men as Rodgers had already made all three substitutions.
Nevertheless, Sturridge found the net against the run of play, latching on to substitute Philippe Coutinho's pass as a poor effort from Bialkowski failed to prevent his shot from dribbling over the line.
Shortly into the second extra time session, Henderson - who had come on for the injured Joe Allen - burst through from midfield before nutmegging Gary Liddle and side-footing home to round the scoring out.