Sunderland skipper Lee Cattermole returned to the starting XI after suspension. Right-back Phil Bardsley played after recovering from illness and sore ribs to play from the start, and midfielder Sebastian Larsson also returned after being rested against QPR at the weekend.
Sunderland made the early running without ever looking like they would break down a well-organised Everton defence.
The Toffees, by contrast, looked sharp on the break and dangerous from set pieces. Sunderland goalkeeper Simon Mignolet was forced into three good saves in a matter of moments around the 22-minute mark.
First, the Belgian number one brilliantly blocked a header from Tim Cahill from Gibson's corner from the left. Then he again did well to keep out Leon Osman’s attempt at the far post to deflect Marouane Fellaini's deep cross goalwards. A minute later, the 23-year-old again denied Cahill, this time from a Gueye corner.
Everton needed to score in this purple patch of play, and they did in fine style in the 24th minute. Fellaini released Gueye down the left and the Frenchman got past Bardsley before crossing low for January signing Jelavic to shoot home low past Mignolet.
Sunderland manager Martin O’Neill will perhaps question the defending, as Sotiros Kyrgiakos was out of position in the centre and allowed the Croat a run at the ball.
Compared to the fluency that Everton were showing, Sunderland’s attacking threat was a great deal more ponderous. Stephane Sessegnon was putting himself about with vigour, and James McClean occasionally looked dangerous, but Nicklas Bendtner in particular struggled to get into the game.
The Black Cats’ best chance of the half fell to the Dane in the 32nd minute, however. McClean’s dangerous cross from the left dropped to the on-loan Arsenal striker at the second attempt, but Leighton Baines was alert and brave to throw himself at the ball and make a last-gasp block.
The second half mirrored the first, with Everton again looking the more dangerous.
First, Osman almost curled a shot into the top corner with a volley after Gueye's corner had been cleared.
Then the Toffees scored their second goal three minutes before the hour.
Vaughan, on for the poor Kyrgiakos, was the culprit. Fellaini overhit a pass across goal to Jelavic, but, even though off balance, the Croat managed to get a weak shot on goal. Vaughan got back to cover, but then in a whir of legs managed only to divert the ball into his own net before falling to the floor in horror.
Four minutes later, Jelavic should have wrapped up the tie, but shot wide after being put through one-on-one with Mignolet by Cahill’s pass. The chance came from another Sunderland error, with Michael Turner giving the ball away to Cahill under pressure.
Mignolet then blocked again from Jelavic in the 65th minute, after the former Rangers player had been set up by a Baines cutback.
The game descended into a succession of niggly tackles, with referee Lee Probert showing five yellow cards to Everton players and three to Sunderland.
O’Neill’s side were denied a consolation in injury-time, with an off-balance Frazier Campbell seeing a shot blocked by Howard’s legs.