It was Cooper who started it off early in the first half with a magnificent 20-yard strike to celebrate his 31st birthday, while Cox hit his first goal for the club and grabbed a second after the interval.
The Bluebirds were without the suspended Jason Koumas and the injured Steve Thompson, while the Potters were missing Sammy Bangoura who had failed to return on time from the African Cup of Nations.
Cameron Jerome up front with Nsungu slashed a shot across the Stoke goalmouth as City began on the attack.
Nsungu was booked in the ninth minute after a high tackle on Lewis Buxton and two minutes later a sweeping move by the visitors saw new loan signing Josip Skoko head wide of Neil Alexander's post.
A mistake by Chris Barker gave former Swansea striker Mamady Sidibe an opening but the full-back recovered to give away a corner.
Skoko was back in the action in the 15th minute when he just failed to meet a Peter Sweeney cross when well placed in front of the home goal.
Cardiff went ahead in the 18th minute with a superb shot from Cooper. Receiving the ball down the right flank, he cut inside before unleashing a rocket into the top left-hand corner of Steve Simonsen's net.
On the half hour Cox doubled the lead from a Cooper corner. Jerome flicked the ball on and it ran on to the unmarked Cox who drilled it low into the net.
Three minutes before the interval Sidibe went close with a header from a Sweeney free-kick but that was a rare attack after City had dominated the opening period.
A flashing drive by Willie Boland in the 48th minute needed a fingertip push over by Simonsen.
Darren Purse almost increased the lead when he swung at a loose ball following a Cooper corner only to strike the angle of the post with Simonsen well beaten.
Cox netted his second of the afternoon in the 68th minute when he glanced a header from a Neal Ardley free-kick into the far corner of the net.
The Potters used all three subs in an effort to gain some consolation and while they mounted a number of raids, City's defence held firm for a deserved victory.