Liverpool got lucky on 68 minutes when Jordi Amat slid in ahead of Henderson to try and clear but the ball ricocheted back off the England international and looped over Swansea keeper Lukasz Fabianski.
Victory took in-form Liverpool to within two points of fourth-placed Manchester United, who they host on Sunday. Chelsea lead on 64, ahead of Manchester City (58) and then Arsenal (57) and United (56).
Since losing to United at Old Trafford in mid-December, a defeat which left Liverpool languishing in 11th place, Brendan Rodgers' side have won 10 and drawn three in the league.
"It was a great pass from Daniel (Sturridge). It was a little bit fortunate, but you've got to be in the right positions to score," Henderson, who scored for a third consecutive league game, told Sky Sports.
"It's a great result. We were very disappointed with our first half. The gaffer had a few words to say. I felt we passed the ball a lot better in the second and were solid at the back."
Liverpool were indebted to goalkeeper Simon Mignolet for keeping out Swansea in a first half shaded by the Welsh side and Rodgers praised the Belgian international, who was dropped for poor form earlier in the season.
"There's no doubt Simon Mignolet's back to his best. We needed him in the first half. He's been excellent. Clearly now, you see his confidence, along with the team's, he said.
Mignolet made a fine-one-handed save to deny Swansea striker Bafetimbi Gomis who was cleared to play following his collapse at Tottenham Hotspur in his side's last league game on March 4.
He was also equal to Gylfi Sigurdsson's curling effort but Liverpool improved after the break.
Phillipe Coutinho forced Fabianski to keep out his low shot from Raheem Sterling's cutback and after Henderson had scored Swansea never seriously threatened to equalise.
Sturridge went close to a second goal late on but rolled a shot against the post in stoppage time.
