The hosts got off to the perfect start as Sturridge latched onto Raheem Sterling's through ball before rounding Michel Vorm and Henderson added a sublime second, curling into the top corner after 20 minutes.
Liverpool appeared to be cruising to a third consecutive Premier League win, but former midfielder Jonjo Shelvey pulled the visitors back into it with a stunning first-time strike which all corners of Anfield stood to applaud.
The home crowd was then left stunned four minutes later as the Swans completed their comeback thanks to Wilfried Bony's deflected header, but the hosts were soon back in front when Sturridge nodded home Luis Suarez's cross.
Swansea were again level just minutes into the second half as Bony calmly converted from the spot after he was bundled over inside the area by Martin Skrtel.
Both sides had numerous chances to take the lead but it was the hosts who did so as Henderson grabbed his second, slotting home from close-range after Vorm had parried Suarez's strike kindly into his path.
It was a breathless afternoon at Anfield and the result leaves Brendan Rodgers' side just four points off the top.
The frenetic pace was set in just the third minute, when the hosts scored a wonderful opener on the break.
Sterling bent a ball into Sturridge's path off the outside of the boot, the striker still needed 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 before they doubled their lead against the run of play.
Sturridge 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.
However, 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 inside.
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 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 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 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 and the striker 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.
Liverpool eventually regained their lead as Henderson doubled his season tally in a day, slotting home after Gerrard's strike squirmed away from Vorm and into his feet.
Swansea threatened to equalise in the closing stages, but the Reds went even closer as Gerrard's deflected effort crashed off the post.