Wade Elliott's early goal sealed a 1-0 win over the Blades at Wembley.
The result means the Clarets must pay Scunthorpe £200,000 as part of their signing of forward Martin Paterson from Glanford Park last summer.
Paterson was not the only former Iron in their side, with Graham Alexander tasting victory at the ground he lost out with with Scunthorpe in the 1992 Division Four final.