The hosts started well enough but fell behind with Falcons' firstattack when a blind-side break from a line-out allowed Luke Fielden to feed Hogg, who charged in from 15 yards.
And they never really recovered from the loss of two wickets in the first six overs against a disciplined Derbyshire attack missing four front-line bowlers.
Although Sol Bamba moved into midfield to make way for returning centre-half Chris Hogg, a first starting place for former Gretna playmaker Fabian Yantorno meant it was a particularly attack-minded Hibs line-up.
United almost took the lead with their first meaningful attack in the 21st minute, when Jon Daly's bullet header was cleared on the line by the alert Rankin.
Kelleher was denied on the line by a timely Martyn Williams tackle after the visitors' best period of attack midway through the first half, but the Blues had no such opportunity.