Sturridge restored Liverpool's lead before the break, but Swansea striker Bony scored from the penalty spot two minutes after the restart to make it 3-3.
The winning goal eventually arrived in the 74th minute courtesy of Henderson, who found the target from close range to seal all three points for the hosts.
Below, Sports Mole has looked back over the Premier League match to determine if the result was a fair one.
On target: 10
On target: 5
Was the result fair?
Swansea will have trudged back to the away dressing room at Anfield wondering how they have not claimed at least a share of the spoils. You can understand their frustration because they created a number of good openings. The difference, though, was the quality of players that Liverpool can call upon in the final third of the pitch. They made the most of their opportunities and for that reason should be regarded as the deserving victors, just.
It was a display that showed just why Liverpool are likely to finish in the top four, but also why they are probably not going to win the Premier League. They can score goals for fun, particularly against teams with leaky defences like Swansea. However, manager Brendan Rodgers has defensive problems of his own. The likes of Steven Gerrard and Henderson were often over-run in midfield, which piled pressure on to the back. If they continue to perform like that, then better teams with a tighter rearguard will exploit it. Still, three points are all that matter and the Reds secured them.
There was very little difference between the performance that Liverpool put in and the one that Swansea produced. Quite simply, the Swans were just outscored. While manager Garry Monk will no doubt be unhappy to see his side concede four goals, he will know that they will not come up against the likes of Luis Suarez and Sturridge every week. Bony's stock also continues to rise following another performance where his power and pace caused problems.
Sports Mole's man of the match
Jordan Henderson: He's not known for his exploits in front of goal, but the midfielder ended up winning Liverpool the match. His first goal was a curling effort from the edge of the box, while his second showed real tenacity not to give in. The goals aside, he also kept possession well and made numerous tackles.
Rodgers may need to have word with Martin Skrtel. He has been seen on a handful of occasions pulling back strikers inside the area this season, but up until now, he had been getting away with it. Not today, though, as he needlessly hauled down Bony and was quite rightly penalised. It allowed Swansea's Ivorian striker to level the score at three goals apiece from the penalty spot and had Henderson not struck late on, Skrtel's error would have cost his team two points.
As mentioned above, Michael Jones was correct to award a penalty against Skrtel. It's the sort of offence that often goes unpunished, so that was pleasing to see. Apart from that, it was a largely uneventful afternoon's work for the man with the whistle.
Liverpool: Next Saturday, Liverpool will make the long trip to the St Mary's Stadium to face Southampton.
Swansea: Having played out a 0-0 draw with Napoli during last week's Europa League first knockout round opening leg, the Swans travel to Italy on Thursday for the return meeting.