Neill Collins’ side broke the deadlock in the 21st minute when Liam Roberts fizzed a ball into John McAtee, who knocked it behind for Cole to curl home.
Burton had to wait 42 minutes for their first chance of the game. After a defensive mix-up, Thomas Hamer had space to shoot inside the box, but Roberts dived to make the save.
Barnsley came close to doubling their lead in the 54th minute. McAtee passed to Cadden on the left, who pulled back for Herbie Kane, but his effort was blocked.
Burton had an opportunity to pull one back 10 minutes later when Beryly Lubala found Mark Helm inside the box, but his low strike was blocked well by Jamie McCart.
The Tykes grabbed their second in the 68th minute. Adam Phillips snuck a ball through to substitute Corey O’Keeffe on the right, who swept in a low ball for Cole to slot home.