Stoke looked so comfortable for so long, it seemed only a matter of time before a second goal came.
Instead, they were stung by a bolt from the blue when Eric Bocat reacted to a loose ball by getting in an unnecessary tangle with Mohamed Belloumi in a seemingly harmless corner of the penalty area.
Belloumi went down and referee Ben Toner pointed to the spot.
Johansson dived to his right to save Palmer's penalty, but the former Coventry man was first to react to the rebound and headed in.
If Stoke felt aggrieved about Hull getting back on level terms, it was about to get a whole lot worse for the hosts.
Wilmot’s slip freed the Tigers to attack and some fine trickery from Slater gave him an opening to fire his side in front.
The turnaround was then complete when Wilmot’s own goal trickled through Johansson’s legs and just over the white line to earn Hull their first win since 24 April, when they won, also in the Midlands, at Coventry.