Goals from Wes York (2), Louis Moult (2), Andy Bishop and Steve Tomassen saw the Dragons comfortably home, even with Joe Clarke having to play the full second half in goal for the injured Dan Bachmann.
Wrexham started with the team which had finished the second half with the upper hand on Saturday and immediately took control.
With Stockport's five at the back sitting deep and surrendering the flanks, Wrexham controlled possession and, after about a quarter of an hour, started to turn pressure into chances.
Louis Moult shot wide from the edge of the area and a goalbound Connor Jennings shot hit Andy Bishop before the breakthrough finally arrived.
But it was on 20 minutes that the game started to swing Wrexham’s way and it was in spectacular fashion that we forged in front.
County had been defending in numbers when Wrexham found their way to goal blocked by a wall of blue shirts and we eventually had to settle for another corner after Mark Carrington’s effort from the ‘D’ spun away off a defender.
Carra trotted over to the left to take the flag kick himself and with plenty of bodies in the box, elected to pull his pass back to Wes York on the corner of the box. The winger took one touch and launched a thing of beauty over the keeper and into the back of the net from 25-yards.
Within a couple of minutes it was 2-0 when York got round the back of the defence and pulled the ball across to Moult, who stretched to sidefoot the ball home.
While it was quite a straight forward finish, it actually came at the end of slick flowing move that involved no less than 14 passes between red shirted players!
Soon Andy Bishop had a chance to make it 3-0 when a quick Robbie Evans free kick put him one-on-one with the Stockport keeper but Danny Hurst followed up his string of saves on Saturday with a fine parry.
County began to claw themselves back into the match and two minutes before the break when Chris Churchman rode a tackle from Robbie Evans to shoot past Dan Bachmann from outside of the box.
Then Hurst was back in action, making a fine one-handed save to deny Bishop again in the first minute of added time, and when Bachmann failed to emerge for the second half it looked like Hurst's save might turn out to be crucial.
Joe Clarke donned the gloves and almost immediately had to make a routine save. Fortunately for Wrexham he wasn't tested again as the Reds took complete control of the second half.
Three minutes after the break it was 3-1, when York's cross from the right picked out Bishop, who finally got the better of Hurst with a close range header.
Despite the loss of Harris with a worsening virus, Wrexham continued to dominate, and York snatched his second of the match when Carrington, shifted into the centre of the pitch to impressive effect following Harris' departure, lifted a fine ball over the top, allowing York to run clear and flick the ball neatly over Hurst.
Moult made it five after Bishop did well to regain possession and clip a fine ball over the Stockport defence, withstanding pressure from a defender behind him to coolly slot the ball under the onrushing Hurst, and Steve Tomassen rounded off the scoring when he got onto the end of York's third assist of the night and finished in a similar manner to York's second goal.
Both Moult and York would have chances to go home with the match ball: York pulled a shot from the edge of the area narrowly wide and Moult was disappointed not to finish when he ran into the box late on and hit the sidenetting with his weaker foot.
However, the team did more than enough to qualify for the next round.
Wrexham: Bachmann (Clarke 46), Carrington, Smith, White, Ashton; Harris (Tomassen 52), Evans (Keates 65), York, Jennings; Moult, Bishop.
Subs: Bailey-Jones, Rushton.
Stockport: Hurst, Todd (Russell 58), Duxbury, O’Halloran, Belezika, Fagbola, Lees, Churchman, Sharp (Spencer 65), Dennis, Ryan.
Subs: Ormson, Lofthouse, Hancock.
Referee: Karl Evans (Lancs)
Wrexham: Evans (57, foul)
Stockport: Churchman (90+2, foul)