The deadlock was broken in just the third minute, when the hosts scored a wonderful goal 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.
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.
Shelvey matched Henderson’s finish, stroking the ball into top corner from a similar position.
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.
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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.
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.
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.
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.