What proved to be Akranes' winning goal came just one minute into the second half when Julluson netted his second of the night.
Paul Peschisolido came closed to equalising for The Rams but his tame header from close-range never troubled the home side's keeper.
Derby: Grant, Kenna, Vincent (Jackson 60), Davies, Johnson, Bolder, Bisgaard, Idiakez, Rasiak (Tudgay 60), Peschisolido (Doyle 60), Smith.