As the Bluebirds upped the tempo, Darren Purse headed another Koumas cross just wide of Boaz Myhill's right post.
Cardiff struck in the 22nd minute with a brilliant piece of opportunism from Jerome. Purse headed on and as two Hull City defenders and the keeper waited for the ball, Jerome nipped in to chip into the empty net.
Two minutes later, Hull missed a glorious chance to level. Jon Parkin raced clear of Purse and had two efforts at blasting the ball into the net from point-blank range but somehow Neil Alexander kept his shots out.
He was the busiest of the two keepers and was forced to dive to his left to push out a shot from Kevin Ellison on the stroke of half time.
Hull pushed forward in the second half and Taylor threw caution to the wind in the 63rd minute by bringing on his remaining two subs. The changes had an immediate effect, with Purse forced to head away a cross from Stuart Green.
Hull were now on top and Alexander was injured as he went down low to save at Parkin's feet.
In a surprise switch, Dave Jones took off Jerome and brought on Rhys Weston to bolster the defence but the visitors still looked dangerous. The Bluebirds held on thanks to a series of saves from Alexander.