Huddersfield were the better side in the first period but poor finishing cost them the chance to take an early lead.
Lee Novak was particularly guilty of wasting opportunities, firing over from 12 yards despite having plenty of space inside the Millwall box.
Jermaine Beckford was close to replying almost immediately but his curled effort from the edge of the box was wide of the target.
Millwall emerged from the half-time break as the more impressive team and it was Wood who managed to double his side's lead.
Henry added a third with 10 minutes remaining by striking a low shot past a helpless Smithies in the Huddersfield goal before the visitors had Jack Hunt sent off for dissent.
Liam Trotter completed an emphatic victory for the Lions with a superb curling effort from just outside the area.