Having seen Paul Benson spurn a half chance when his low shot from the edge of the six-yard box was safely gathered by Lee Harrison in the home goal, the Bees began to enjoy the better of the early chances.
Puncheon, who was the Bees' biggest threat, picked out Michael Leary who fired a matter of inches past the near post.
Liam Hatch then had a better chance to open the scoring, but, having been put clean through from Nicky Nicolau's superb through ball, he horribly miscued at the vital moment and saw his effort bobble wide.
Barnet did though take the lead in the 40th minute from the penalty spot. Puncheon won the kick having been brought down in the box by Marlon Patterson and the midfielder stepped up to coolly convert, sending Tony Roberts the wrong way.
Dagenham came within inches of levelling the score before the break, but from Sam Saunders' corner, Benson headed against the bar, before deep into stoppage time the Bees were reduced to ten men when captain Ismail Yakubu received his marching orders for picking up a second yellow card.
However, the numerical disadvantage didn't prevent Barnet doubling their lead five minutes after the restart when a swift counter-attack saw Hatch and Adam Birchall link up to play in Puncheon, he kept his composure to fire in his second of the match.
Barnet looked the most threatening throughout the second half and Puncheon completed his treble with a stunning strike into the top corner.
The Daggers pulled one back two minutes from time when from Sam Sloma's free-kick Shane Huke fired the ball past Harrison, before substitute Giuliano Grazioli hit the post for Barnet at the death.