The Swans twice tied the match but could not stop Brendan Rodgers' men from taking all three points in an entertaining contest at Anfield.
Daniel Sturridge scored twice to become only the second player — after former Manchester United forward Ruud van Nistelrooy — to net in eight consecutive Premier League fixtures. But the England striker's brace was only one part of a remarkable story that unfolded, with five first-half goals setting the tone for a top-flight classic.
It took just three minutes for Sturridge to break the deadlock and Henderson's pinpoint finish put Brendan Rodgers' side in the ascendancy midway through the opening half.
The game was turned on its head as Jonjo Shelvey and Wilfried Bony scored within three minutes of each other, the former drawing applause from the fans of his former club after arrowing a stunning first-time effort into the top-right corner.
Sturridge netted again before the break to take his tally for the season to 18 league goals, but Bony grabbed his second from the spot early in the second half.
It was Henderson who had the last say, though, his close-range finish sealing the points 16 minutes from time.
Liverpool signalled its intent from the start and took the lead as Raheem Sterling provided a defense-splitting pass for Sturridge, who deftly side-stepped Michel Vorm before slotting home.
The visitors did show some early promise, but Henderson's superb right-footed strike from the edge of the box seemingly put the hosts in control with 20 minutes gone.
Swansea needed a response and it was provided two minutes later in emphatic fashion by Shelvey, who scored one and set up another in September's reverse fixture, but was also responsible for a dreadful mistake that led to Sturridge's equalizer in a game that ended 2-2.
He exorcised those demons with a glorious shot from 20 yards that left Simon Mignolet with no chance. The 21-year-old, making his return to league action from a hamstring injury, refused to celebrate but was applauded by many of the home contingent, whose charitable nature was less apparent three minutes later when Bony leveled matters.
Shelvey was on the receiving end of a hefty challenge from Martin Skrtel and the Liverpool defender then appeared to get a decisive touch on Jonathan de Guzman's subsequent free kick, which had been headed goalward by Bony, wrong-footing Mignolet.
The breathless pace of the game continued and, remarkably, Rodgers' men were ahead again before the interval, Sturridge doubling his tally with a neat back-post header from Luis Suarez's cross.
Suarez almost joined the fun himself, lobbing an effort from the halfway line just wide of Vorm's goal.
Swansea tied the match two minutes into the second half after Skrtel, already on a yellow, clumsily felled Bony, with the Ivorian picking himself up to convert the spot kick.
Yet just as it appeared as though City boss Garry Monk would earn a point against his former manager, as Suarez's stinging effort was turned into the path of Henderson to poke home from inside the 6-yard box.
The result moves Liverpool, which hit the post late through Steven Gerrard, back to within four points of first-place Chelsea, while Swansea remains four points above the drop zone having won just two of its last 13 league games.