In-form striker Sturridge netted his second of the game in the 105th minute before Henderson sealed the Premier League outfit's progress to round three five minutes later by making it 4-2 with a cool finish.
Those goals came after second-half strikes from Yoann Arquin and Adam Coombes had brought their spirited League One visitors level, following Raheem Sterling's early opener and Sturridge's first before the break.
Brendan Rodgers named a strong team despite the prospect of a Premier League fixture against Manchester United on Sunday, with captain Steven Gerrard and Sturridge both starting, along with the likes of Glen Johnson and Kolo Toure.
Meanwhile, Notts County boss Chris Kiwomya made two changes to the team that lost to Stevenage on Saturday, with Jamal Campbell-Ryce and Aston Villa loanee Enda Stevens coming into the side.
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’ 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 backline and the 23-year-old striker 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 Sturridge, 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, as he lashed a left-footed shot off the post after Sterling had found him with a well-weighted pass.
However, Kiwomya’s side responded admirably after the interval and their spell of pressure paid dividends around the hour mark when Arquin pulled one back.
Manny Smith rose unmarked to head a free-kick back across goal and the Martinique international beat Toure and Andre Wisdom to power a header home.
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.
And the home side were made to pay for their profligacy when Coombes equalised after good work from Arquin to force extra time.
Notts County began the additional 30 minutes the brighter of the two sides and their chances were boosted further when Toure was carried off with a groin injury, leaving Liverpool with 10 men with Rodgers having already made all three substitutions.
Nevertheless, Sturridge then found the net against the run of play, as he latched on to substitute Philippe Coutinho's pass before Henderson - who had come on for the injured Joe Allen - burst through from midfield before nutmegging Gary Liddle and side-footing home.
