The Reds, who are currently undefeated in the Premier League since the turn of the year, almost opened their account inside the opening 30 seconds after Daniel Sturridge exchanged passes with Raheem Sterling and Adam Lallana before forcing Tom Heaton into an early save.
Liverpool continued to press for an early breakthrough as Philippe Coutinho's corner reached the head of Henderson, but the stand-in captain was unable to keep his effort down.
Henderson again threatened an opening goal in the 10th minute after Heaton's attempt to punch Coutinho's free kick saw the ball land at the midfielder's feet, only for the skipper to sky his shot.
Sturridge almost broke the deadlock as Burnley failed to clear a corner, when Lallana connected with Emre Can's pass and the England forward reacted quickly as he steered the ball just wide of the Clarets goal
However, Liverpool wouldn't be denied for long and it was Henderson who followed up his wonder-strike in the victory over Manchester City by volleying into the corner of the net from the edge of the box, after Coutinho's initial effort was blocked.
Brendan Rodgers's side almost doubled their advantage, moments before the break, after Coutinho's slide-rule pass found Sturridge, who only had Heaton to beat, but the Burnley keeper was able to keep out the England international.