Hibernian loanee Murray delightfully curled home from distance.
Leading the play-off 2-0 following Tuesday’s first leg, the Citizens saw their advantage wiped outside inside 20 minutes at Galabank following Tony Wallace’s early penalty and a Tommy Goss header.
But Alan Maybury’s side were not to be denied as Hibernian loanee Murray – who opened the scoring in the first meeting – delightfully curled home from distance to claim the decisive goal eight minutes into the second period.
Annan finished seven points above their opponents in the regular Liga to årstid.
Striker Rob McCartney hit the side-netting in the closing stages as the hosts searched for a way back into the tie, while Edinburgh goalkeeper Brian Schwake made a string of fine saves to deny them.