The Clarets broke their transfer record to sign the New Zealand worldwide from the Whites during the summer and the striker has hit the ground running since stepping up to the Premier League, netting twice in four appearances.
Leeds eliminated the Premier League club during a penalty shoot-out at Turf Moor, holding their nerve having twice lost a one-goal lead late in normal time.
The result was a shot in the arm for United and their head coach after Millwall inflicted the Championship leaders' first defeat of the season on Saturday.
"There was too much (drama) in the last 13 minutes but that's football".
After a largely tiresome extra time it went to penalties.
Here's how Twitter reacted to Leeds United cup victory...
But just as time looked up, a stunning free-kick from Robbie Brady made for extraordinary scenes, as it was 2-2, and extra time did, indeed come.
Inevitably the tempo dropped as both sides tried to regain their breath, Leeds opting for the fourth substitution permitted in extra-time to try and keep the Clarets at bay.
Wood came on as a second half substitute and scored from the spot to tie the game up in the 89th minute.
Burnley manager Sean Dyche gives starts to Phil Bardsley, Kevin Long, Jeff Hendrick, Ashley Westwood, Ashley Barnes and Sam Vokes.
Leeds: Lonergan, Borthwick-Jackson (Sacko 59), Berardi, Shaughnessy, Ayling, Vieira, Klich, Dallas, Roofe (Alioski 104), Cibicki (Hernandez 79), Grot (Lasogga 85).