The Croatia international fired the visitors ahead in their quarter-final replay with 24 minutes gone, but then passed up a series of opportunities to add to his tally on a night when the men from Merseyside were more than a match for their hosts.
It took a 56th-minute own goal from Black Cats substitute David Vaughan to ease David Moyes' men further clear, and although the Wearsiders battled all the way to the whistle in front of a boisterous crowd of 43,140 at the Stadium of Light, there was no way back.
Everton simply had too much for their hosts with Marouane Fellaini, Leon Osman and Magaye Gueye outstanding in the middle of the field as they set up a chance to gain revenge for their derby defeat earlier this month.
The visitors only arrived an hour before kick-off due to heavy traffic, but they showed few ill effects as they set about the task of extending their extraordinary recent dominance over the Black Cats.
They had lost none of the previous 16 meetings between the two sides in all competition before kick-off, and the chances of them doing so tonight looked remote throughout.
After Jelavic had gone close, Sunderland goalkeeper Simon Mignolet made an eighth-minute save at his near post to deny nemesis Tim Cahill yet another goal against them.
The Belgium international was called upon twice in quick succession as the clock reached the 20th minute with Cahill forcing a fine reaction saved from Darron Gibson's left-wing cross before Jelavic tested him with another close-range header.
When the opening goal finally arrived four minutes later, it was no surprise that it did so at that end of the stadium as Sunderland were undone by Everton's pace and accuracy of passing.
Fellaini slid the ball into the run of Gueye down the left, and his firm cross was swept confidently past Mignolet by Jelavic.
Sebastian Larsson was unfortunate not to control Stephane Sessegnon's inventive 28th-minute ball over the top, and the Benin international played striker Nicklas Bendtner in seconds later only for an offside flag to make his effort, which was saved by Tim Howard in any case, irrelevant.
Leighton Baines had to clear hurriedly from Bendtner in front of his own goal four minutes later after Sylvain Distin had failed to cut out James McClean's cross, and full-back Phil Bardsley sent a long-range volley over the top two minutes later after keeper Tim Howard had punched McClean's corner clear.
Larsson curled a 35th-minute free-kick a yard wide, but Howard remained largely untroubled and as the men ahead of him continued to threaten on the break, Everton looked capable of adding to their tally.
They might have done so twice within the opening five minutes of the second half.
Osman sent a viciously-swerving right-foot volley inches wide after Craig Gardner and Michael Turner could only help on Gueye's 47th-minute corner, then Lee Cattermole had to block Gibson's side-footed effort three minutes later with Gueye the provider once again.
But it was 2-0 11 minutes after the restart when Fellaini broke free and squared for Jelavic, and although the pass took him wide, his ball back across goal was turned home by the unfortunate Vaughan, whose attempted left-foot clearance hit his left and rolled into the net.
Sessegnon hit the angle of bar and post from Larsson's deep corner on the hour, but Jelavic shot wide with only Mignolet to beat seconds later as the visitors scented blood.
Jelavic could have added to his tally with 66 minutes gone, but his snapshot was palmed away by Mignolet, and the hugely impressive Gueye shot just over a minute later with the home rearguard again in tatters.
Sunderland, with striker Fraizer Campbell added to the mix, battled manfully for a way back into the game until referee Lee Probert finally drew proceedings to a conclusion, and although Howard won a one-on-one battle with Campbell in injury-time, they never looked like finding one and were ultimately well beaten on the night.