The initial tie on March 17 ended 1-1 at Goodison Park, forcing a replay, and it was the Toffees who came through to book a Wembley date where they will meet Merseyside rivals Liverpool on April 14.
Everton were providing plenty of problems for the Black Cats, mainly through Jelavic, but it was an own-goal which gave them a second when substitute David Vaughan unfortunately turned in the former Rangers man's cross.
Sunderland keeper Simon Mignolet came up with a number of vital saves to deny the Toffees in the first half whilst his Everton counterpart Tim Howard was mainly untroubled except for when Stephane Sessegnon hit the corner of the crossbar from Seb Larsson's cross.
Everton could have added a third on a few occasions later in the piece with Mignolet again denying Jelavic before Gueye shot over the bar but the damage had been done as David Moyes guided his side into the final four where they will re-live the recent 3-0 derby loss at the hands of the Reds.
Stadium of Light, Sunderland - 43,140