Mame Biram Diouf opened the scoring at the Racecourse ground, racing onto Glenn Whelan’s pass before sliding the ball under Cameron Belford and into the back of the net.
The Premier League side doubled their advantage shortly afterwards as Jonathan Walters latched onto Erik Pieters’ free-kick and looped the ball over the Wrexham keeper.
Stoke scored their third just four minutes before the break as Marco van Ginkel burst into the Wrexham half to fire past a helpless Belford.
Mark Hughes made two changes at the break as Mason Watkins and Joselu replaced Pieters and Peter Crouch respectively, but the onslaught did not stop as the visitors continued to pile on the pressure.
They had their reward in the 62nd minute when Walters grabbed his second of the night with a simple tap-in.
Joslelu added a fifth just two minutes later as his deflected shot handed the Spaniard his first goal in Stoke colours.
Wrexham had an opportunity to pull one back when Marc Wilson was adjudged to have used his hands in the box but Daniel Bachmann, who was on loan at the Racecourse Gorund last season, saved Kayden Jackson’s penalty.
Stoke rounded of a satisfying night when Dom Telford’s low shot found the bottom corner.