Jenkins, who finished with 27 carries for 178 yards, had touchdown runs of 4 and 34 yards in the third quarter, helping the Blue Devils jump out to a 12-0 lead.
Edisto stormed back to tie the game at 12 early in the fourth quarter after scoring runs by Justin Maxwell and Terrell Maxwell.
But Jenkins score gave the lead right back to Williston-Elko, which held for the victory.
The Blue Devils are coming off another deep playoff run. They reached the SCHSL Division II state semifinals before falling to Christ Church last season.