Sturridge's first came in the sixth minute, before the striker provided the assist for Henderson to double Liverpool's F.C. lead in the 20th from the edge of the area.
Jonjo Shlevey pulled one back for Swansea against his former club three minutes later with a long-range strike before Wilfried Bony headed in Jonathan De Guzman's free kick in the 27th for 2-2.
Sturridge headed in his second to give Liverpool F.C. a 3-2 half time lead, but Swansea levelled soon after the interval when Bony scored a penalty after being fouled by Martin Skrtel.
Henderson then scored the winner in the 74th, converting a rebound from his shot.