Henderson found the net for the first time since November and Tottenham loanee Harry Kane grabbed his first goals for the club as they dispatched the Lions saw off their League Two opponents at the second attempt.
Victory came as a huge relief to Kenny Jackett, who had seen his side score just once in their previous eight outings and thumped 6-0 by Birmingham just three days ago to leave them embroiled in the Championship relegation battle.
It also meant Millwall made it through to the fourth round - and a home tie with Southampton - for only the third time in eight attempts since they lost the 2004 final to Manchester United.
The Lions were eager to bounce back from Saturday's mauling and went ahead after just seven minutes courtesy of Henderson, who had not started the last four games due to injury.
Full-back Jack Smith's long ball was nodded down by Liam Trotter to Kane, who scuffed an ambitious shot straight to the feet of Henderson on the edge of the area.
The big striker controlled the ball, turned and curled a fine effort past Daggers keeper Chris Lewington.
They should have been two up five minutes later when Henderson teed up Liam Feeney in the area but the winger made a mess of his shot.
Teenager Kane was proving a lively presence in the Lions' attack and had two efforts saved by Lewington in quick succession.
But Lewington could do nothing four minutes before the interval when Trotter strode forward before sliding the ball to Kane, who struck a superb low shot from 25 yards into the corner of the net.
The plucky Daggers had chances to pull one back after the restart but Jon Nurse blazed over, Dominic Green drove wide and Millwall were indebted to a brave goal-line clearance by skipper Paul Robinson from Billy Bingham's cross.
But the hosts gave themselves some breathing space 59th minute when Trotter's square pass left Henderson with an open goal.
Four minutes later Richard Rose upended Kane in the area and Henderson rolled in the penalty to complete his treble for his 12th goal of the season.
And Kane completed the scoring in the 65th minute with his second, a header from Henderson's cross, to put a smile back on Jackett's face.