Brendan Rodgers' side continue to be plagued by a porous defence but moved within four points of Barclays Premier League leaders Chelsea after emerging victorious from one of the games of the season.
Henderson had the final word in his side's 4-3 success, forcing home from point-blank range in the 74th minute after his initial effort had been saved.
However, Rodgers' former team played their part with a stirring away performance capped by Jonjo Shelvey's first-time strike and a brace from Wilfried Bony.
The frenetic pace was set in just the third minute, when the hosts scored a wonderful opener on the break.
Raheem Sterling played creator when he robbed Angel Rangel inside his own half and executed an inch-perfect 30-yard pass for the advancing Sturridge.
Sterling's ball, bending into the striker's path off the outside of the boot, still required Sturridge to round Michel Vorm and he did so with aplomb before locating the open net.
Liverpool nearly caught Swansea off guard moments later when Sterling's curled effort was beaten away by Vorm at the near post.
The hosts doubled their lead against the run of play after 21 minutes.
Sturridge's flash of pace - breezing past Neil Taylor as he cut in from the right wing - created the space, but the finish was even better. Henderson, lurking on the edge of the area, controlled the ball with his left boot, eyed up the top corner and dispatched a precision finish with his right.
Swansea's response was immediate and emphatic.
Just two minutes after conceding they created their own version of Liverpool's second, Nathan Dyer playing the Sturridge role as he drifted iside.
And Shelvey was on hand to match Henderson's finish, stroking the ball into top corner from a similar position.
There was no celebration against his former club and Anfield responded with a warm ovation.
The goals kept coming, with the visitors making it four in the 27th minute and Martin Skrtel unfortunate on two counts.
The Slovakian was harshly adjudged to have fouled Shelvey for the initial free-kick and, when Jonathan De Guzman's delivery was headed on by Bony, the ball deflected past Simon Mignolet via Skrtel's shoulder.
Swansea had the momentum, but once more the Reds proved lethal in the final third as they reclaimed the lead after 36 minutes.
A patient build-up stretched Swansea and when Luis Suarez floated a cross for Sturridge, he rose between Ashley Williams and Wayne Routledge to nod home his 20th goal of the season.
There was no further scoring before the break, though did Luis Suarez see an an audacious strike from 45 yards drift a couple of yards wide with Vorm back-tracking.
Less than three minutes of the second half had passed when Swansea equalised for the second time, Skrtel again cursing his luck as he conceded a questionable penalty as he tangled with Bony.
He was at least spared a second yellow card but Bony made no mistake from the spot.
Attack continued to dominate defence with Suarez denied at the near post after a flowing move, and De Guzman twice going close at the other end.
Referee Mike Jones had decisions to make too, Mignolet close to handling outside the area and Suarez calling unsuccessfully for a penalty following a collision with Jose Canas.
A seventh goal seemed inevitable and Liverpool grabbed it with 16 minutes remaining.
Steven Gerrard produced a wonderful 40-yard pass to Suarez, who controlled on his chest and saw his shot blocked by Williams.
Henderson fired the loose ball at Vorm, who saved smartly but allowed it to squirm out of his grasp as the England midfielder followed up to finish the job from a yard. The two goals he bagged meant the England midfielder doubled his season's tally in a day.
Swansea threatened to equalise for a third time in the closing stages, but the Reds went even closer as Gerrard's deflected effort crashed off the post.