In a match that showcased both their devastating attacking power and worrying defensive frailty, Brendan Rodgers's side raced into a two-goal lead before letting Swansea twice draw level - then secured a result that sees them stay in the title race with Chelsea, Arsenal and Manchester City.
Sturridge opened the scoring in the third minute following brilliant work by Raheem Sterling, who won possession in his own half and played a perfect 50-yard through ball. Sturridge beat Michel Vorm to it, and the England striker finished into the empty net. Sterling could well join his colleague in Brazil this summer.
Henderson doubled the lead on 20 minutes with a sweeping finish into the top corner from the edge of the box, but rather than consolidate the home side defended slackly.
Parity did not last long, as a fine Liverpool team move culminated in Luis Suarez crossing for Sturridge to head his 18th goal in 19 Premier League games this season.
Swansea equalised for the second time early in the second half when Martin Skrtel manhandled Bony and the Ivorian beat Mignolet from the penalty spot.
Liverpool came again, Luis Suarez had a penalty appeal rejected after a theatrical fall, and the game's seventh goal finally came after Ashley Williams blocked a Suarez shot. Vorm parried Henderson's first shot on the rebound but could do nothing about the second.
3' - GOAL: Liverpool 1-0 Swansea - Sturridge races onto Sterling's brilliant through ball and rounds Vorm to open the scoring.
20' - GOAL: Liverpool 2-0 Swansea - Henderson has time to tee himself up before picking out the top left corner of the Swansea net.
23' - GOAL: Liverpool 2-1 Swansea - Quickfire response from Shelvey, who pings in off the bar, then gets appaluse from the home fans for his low-key 'celebration'.
27' - GOAL: Liverpool 2-2 Swansea - Swansea are level as Bony's glancing header takes a wicked deflection off Skrtel and beats Mignolet.
36' - GOAL: Liverpool 3-2 Swansea - Sturridge heads home his second from a Suarez cross after a fine passing move as Swansea are cut to shreds again.
47' - GOAL: Liverpool 3-3 Swansea - Ref Mike Jones spots Skrtel's hands on Bony, and the striker places his penalty into the bottom left corner.
67' - Suarez denied penalty - Swansea put the ball out for an injury, Liverpool refuse to give it back from the throw, and Suarez goes down dramatically after slight contact from Jose Canas. No penalty.
74' - GOAL: Liverpool 4-3 Swansea - Henderson wins it, tucking away the second rebound after Vorm repels his first shot.
Daniel Sturridge has scored a goal every 83 minutes in the Premier League this season. That's the same as Cristiano Ronaldo and better than Lionel Messi (95 minutes)
Sturridge has scored on all but five of his 22 Premier League appearances this season.
Liverpool have scored the most goals in the Premier League (70) and have the two top scorers - Suarez (23) and Sturridge (18)
Can Liverpool really challenge for the title defending like this? Nobody provides entertainment like Brendan Rodgers's men, but such is the way they turn what should be a stroll into a white-knuckle ride it's hard to see them topping the table at the end of the season. It would be the ultimate triumph for attacking football if Liverpool overcame Chelsea's pragmatism - or even the more balanced Manchester City and Arsenal.
LIVERPOOL: Mignolet 6, Johnson 6, Flanagan 7, Skrtel 5, Agger 5, Gerrard 5, Henderson 8, Sterling 7, Coutinho 6, Suarez 7, Sturridge 8. Subs: Toure 6, Moses 6, Allen 6.
SWANSEA: Vorm 7, Rangel 6, Chico Flores 5, Williams 6, Taylor 5, Britton 6, De Guzman 7, Shelvey 7, Dyer 7, Routledge 6, Bony 7. Subs: Canas 6, Hernandez 6, Ngog 6.
MATCH IN A TWEET