But one win in 19 Premier League games has seen the Tigers slide down the table and, although they have managed to stay out of the clutches of the bottom three all season, their form suggests that may not be the case for much longer.
The home side's play had an anxiety and tension about it at the start and the visitors should have capitalised on the space being afforded them in attacking positions.
Hull gradually settled and went close when Craig Fagan crossed for Daniel Cousin, whose looping header was tipped over by keeper Thomas Sorensen.
Both teams mainly took the aerial route into the opposition's box and a knock down gave Richard Garcia a sight of the Stoke goal.
We deserve some credit - Pulis
He chested the ball down but, with more time and space than he thought, he failed to take advantage of the chance by shooting too close to Sorensen.
Stoke repelled those rare chances and always looked the more dangerous attacking side and a swift counter-attack from the visitors ended with Lawrence having a shot deflected narrowly wide off Kamil Zayatte.
Lawrence took the resultant corner from which the ball fell to Fuller, who turned and shot into the Hull goal.
Kilbane almost responded shortly after the break only to see his 15-yard shot palmed round the post by Sorensen.
A disciplined and organised Stoke were now increasingly content to sit on their lead, soak up Hull's pressure and try to catch the home side out on the break or from a set-piece.
We must keep believing - Brown
The gameplan nearly came off when a corner fell to Whelan but his shot to come off the corner of the post and crossbar.
But Hull had no such luck when Lawrence unleashed a powerful angled strike past keeper Boaz Myhill, who could just stand and watch.
Dawson curled a 20-yard free-kick into the top corner for a consolation goal, but it could not stop Stoke earning a momentous win, although Geovanni came agonisingly close to an equaliser with a deflected shot in the final minute of injury time.
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