James McFadden's brace and a strike from Yakubu shattered the Owls' Carling Cup dreams as Everton advanced into round four.
The Toffees sealed the tie in the second half, with Premiership class finishing proving the difference.
Owls boss Brian Laws made five changes from the team that beat Hull in the Championship last time out. Peter Gilbert, Richard Hinds, Jermaine Johnson, Wade Small and Marcus Tudgay were each handed starting shirts as Laws shuffled his pack ahead of the trip to Norwich on Saturday.
Gilbert slotted in at left back while Hinds came into the heart of the SWFC defence. Small started down the left flank with Johnson across the other side of the park. Tudgay partnered former Everton striker Francis Jeffers in attack.
It was Jeffers at the centre of the game's first talking point as the hitman had a half-hearted penalty claim waved away by Premiership referee Rob Styles.
New Owls loan recruit Graham Kavanagh then burst forward and warmed the hands of Stefan Wessels with a rasping drive before Johnson sent an ambitious effort past the EFC keeper's right-hand post.
Jeffers sent Johnson darting in towards goal with a delightful feed that caught the Toffees' defence square but the winger powered his shot high and wide.
The Owls continued to press up to half time and looked comfortable at the back as Everton posed little threat in the final third.
The Merseysiders introduced Phil Neville for the second period and the England international created an instant half chance for Leon Osman, who scuffed a tame effort wide from 12 yards.
Both sides began to up the ante with Wednesday enjoying the upper hand as the home fans stoked the atmosphere.
Johnson sent over a fine cross that was punted out for a corner while at the other end, Phil Jagielka fired a long-range drive down the throat of Grant.
Everton stole into the lead on 59 minutes against the run of play. Recalled record signing Yakubu broke free down the right channel and his low cross was calmly converted by McFadden.
With the Toffees starting to show their Premiership credentials, McFadden executed a sublime chip that Grant saw safely over his crossbar.
Wednesday's reply almost restored parity as the hosts carved three chances in as many minutes.
Graham Kavanagh saw his free kick brush the top of the Toffees' net, substitute Deon Burton thumped a daisycutter wide and Glenn Whelan fizzed a shot past the post from distance.
But Everton clinched their passage into the next round with two late goals. McFadden headed home from close range on 84 minutes and Yakubu threaded past Grant 60 seconds later as the visitors coasted to victory.
