A sublime finish from the Croatian handed the Toffees a thoroughly deserved halftime lead, and any Black Cats fightback was soon relinquished following a calamitous second.
A fine first-half performance from the Toffees was capped by a fine finish from Nikica Jelavic after good work from Magaye Gueye.
Martin O’Neill attempted to rejuvenate his lackluster Sunderland side with a number of attacking changes in the second half, but the substitutions had little effect on the game as the Toffees only extended their lead thanks to a calamitous David Vaughan own goal.
O’Neill made three changes to the side that triumphed over QPR on Saturday. Both Phil Bardsley and Sebastian Larsson were restored to the starting XI after illness had ruled them out of the weekend win, with John O’Shea and Jack Colback missing out.
Meanwhile, skipper Lee Cattermole returned to the team having served his four match ban for a red card picked up against Newcastle, with Vaughan dropping to the bench.
For the visitors, David Moyes also made three changes to the side that recorded a terrific 2-0 win at Swansea on Saturday. In came Gueye for the cup-tied Steven Pienaar, while Maroune Fellaini and Johnny Heitinga were given the nod ahead of Tony Hibbert and Phil Jagielka, having both been rested for the weekend win.
It was the visitors who made the brighter start of the two teams, and they perhaps should have taken an early lead when Leon Osman evaded the Sunderland offside trap before clipping the ball in to an unmarked Jelavic, but the Croatian failed to make sufficient contact with his header.
The early warning sparked the home side into action, yet Everton was still posing a threat from set-pieces, and Simon Mignolet had to be at his absolute best to deny Tim Cahill and then Jelavic.
Despite the Belgian’s early heroics he was helpless to prevent the Toffees taking the lead midway through the first half.
After the visitors had worked the ball down the left, Gueye cut back a low, first time ball to the unmarked Jelavic, who supplied a classy side footed finish to beat Mignolet at his near post, sending nearly 7,000 traveling Merseysiders into raptures.
The remainder of the half was a relatively even contest, with both sides being reduced to long-range efforts which never looked like testing Tim Howard or Mignolet, respectively, as the home side departed the field to a chorus of boos.
The second half began in a similar vein to the first with the visitors looking to extend their lead.
And they came within a whisker of doing so just two minutes in, when Osman peeled off towards the edge of the box from a corner. The ball eventually reached the midfielder via a number of deflections and he fired a sweet, right-footed volley just inches over the crossbar with Mignolet rooted to the spot.
With Moyes’ men dominating, O’Neill introduced Vaughan in an attempt to strengthen the Black Cats’ midfield, but the move backfired horribly.
After Fellaini had played a slide-rule ball through to the unmarked Jelavic, the Croatia international slid the ball under Mignolet, only for the retreating Vaughan to bundle the ball over his own goalline in comical fashion.
The visitors’ second of the evening sparked the contest into life as a full-blooded cup tie ensued, with Stephane Sessegnon hooking the ball against the post from two-yards out prior to Jelavic inexplicably firing wide with only Mignolet to beat.
With the game becoming increasingly stretched, Moyes replaced the impressive Gueye with Jagielka as the Toffees shut up shop and saw out an impressive 2-0 victory.
The win means Sunderland has failed to beat Everton in any of its previous 16 encounters. Liverpool awaits Moyes’ men in the semi-final, while Sunderland will now look to concentrate on the chase for European football.