The National League side gave a start to Junior Morias, who returned to the Hertfordshire side on loan from St Mirren.
The first opportunity fell to the Championship side after five minutes, when an effort from distance by Troy Parrott went wide.
Millwall looked the stronger of the two sides. Kenneth Zohore put a free-kick off target from the edge of the area after 10 minutes, while Parrott dragged an attempt wide after 19 minutes.
The deadlock was almost broken by the EFL side after 26 minutes, when a shot by Burey after 26 minutes from a tight angle was well saved by Boreham Wood goalkeeper Nathan Ashmore.
The 30-year old shotstopper was called into action again minutes later, denying Ryan Leonard.
However, it seemed inevitable that the Boreham Wood back line would be breached, and Millwall took the lead on the half-hour mark. Scott Malone fed the ball through to Zohore, who slotted home effortlessly.
The visitors looked to extend their lead, and Ashmore did well to hold a Leonard shot that took a deflection.
With one goal between the sides at the interval, the non-league side started the second half in a more aggressive manner.
Millwall remained dangerous though, and Jake Cooper almost doubled the lead five minutes after the restart, forcing Ashmore to save with a header after running onto a Malone corner.
Boreham Wood created their best chance of the game on 53 minutes, when Matt Rhead put an effort over the bar from less than 10 yards out.
The visitors again almost doubled their advantage after 63 minutes, when substitute Tom Bradshaw aimed wide after making a driving run into the box.
With 16 minutes of the game remaining, Millwall got the second goal their play threatened. An inswinging corner by Malone found the head of Shaun Hutchinson, who connected powerfully to seemingly seal the tie for the south Londoners.
The Championship side continued to look comfortable, and almost netted a third on 77 minutes, when a powerful drive by Malone was matched by a flying save from Ashmore.
Malone again nearly added to the scoresheet in injury time, but he put a shot following a run into the box wide.