Wood was on target after just three minutes and De Laet doubled the lead as the midway point of the first half approached.
The Foxes had chances to win by a greater margin but in the end two goals were enough as the Championship club took the first step to emulating their quarter-final appearance of last season.
The League Two side made an awful start. A long ball into the box by Paul Konchesky sought out Wood and the former West Brom striker outmuscled Aaron Webster before guiding a right-foot volley into the corner of the net for his third goal in two games for his new club.
John McGrath came close to a quick response for the visitors but his curling effort from the edge of the box just cleared the crossbar.
The Championship club extended their lead after 21 minutes. Leicester showed patience in and around the penalty area before a challenge from a Burton defender saw the ball squirt out to the right-hand side where De Laet fired home from 10 yards out.
Sloppy play from the hosts almost allowed Albion a route back into the game after half an hour. Drury seized on the loose ball and teed up Robbie Weir who was denied by an excellent save from Kasper Schmeichel, diving to his right.
Marko Futacs had chances either side of half-time to put the game beyond Burton, first steering a shot from Matty James' cross against the crossbar and then blasting over after neatly bringing down Andy King's centre.
Calvin Zola showed lovely technique to control Jacques Maghoma's cross with his back to goal but his acrobatic volley was straight at Schmeichel.
McGrath cleared Michael Keane's header off the line after an hour and Leicester threatened a third goal again midway through the second half when King's 20-yarder skidded narrowly wide.
Substitute David Nugent headed over shortly afterwards but in truth the Foxes played much of the second 45 minutes in second gear and comfortably eased through to the next round.