Two goals in six minutes rocked the Championship side as the Welshmen romped into a first-half lead, and Wrexham then booked their place in the second round with two more goals after the break.
A makeshift Wednesday side - hampered by a 13 strong injury list and no fit strikers - had no answer to the League Two side's attacking prowess.
Wrexham, currently sixth in the table and unbeaten, made a bright start and an early free-kick from Mark Jones was blocked by the Wednesday wall.
The Owls briefly threatened through Chris Brunt, but his first effort at goal flew wide, and his second high.
After 24 minutes Brunt finally found the target with a stunning low strike from 30 yards, but Michael Ingham matched it with a full stretch dive to fingertip the shot wide.
Then, against the run of play, Wrexham broke the deadlock. Ex-Doncaster Rovers striker Neil Roberts pounced to stab in a close-range shot from a Mark Jones knockdown after 33 minutes, despite claims of handball from Wednesday.
And when Lee Bullen, back after a three-match ban, was robbed close to the halfway line, Chris Llewellyn raced into the area and rifled a left-foot shot through Chris Adamson's legs to put Wrexham in control.
Wednesday manager Paul Sturrock sent skipper Bullen up front in a desperate second-half bid to boost the home side's firepower, but it was the visitors who found the net again.
The third goal came in the 63rd minute when Matty Done hit a crisp shot to put the game beyond Wednesday.
Glenn Whelan pulled one back for the hosts with a spectacular strike from 25 yards 11 minutes from the end.
But in the 84th minute Mark Jones added Wrexham's fourth with a low strike from the edge of the box.