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, 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.
After Jelavic had gone close early on, 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 then produced a fine reaction save from another Cahill effort before Jelavic tested him with a close-range header.
The opening goal finally arrived four minutes later. Fellaini slid the ball into the run of Gueye down the left, and his firm cross was swept confidently past Mignolet by Jelavic. tephane Sessegnon then 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.
In 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, before 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's snapshot was then 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.
Source: PA
Source: PA