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Hull were eager to assert themselves in the opening minutes, looking to get on the front foot and attempting to target Bardsley at right back.
It was a system Stoke struggled to pick holes in despite their advantage.
There were half chances an Odemwingie dribble into the area, Erik Pieters' wayward drive but the hosts were holding up well.
Lacking the bite that Whelan would have offered, he allowed Tom Huddlestone to breeze past him and unleash a low shot that Asmir Begovic could only palm back out.
Their first warning came after only three minutes when Peter Odemwingie played in Mame Diouf.
Jelavic anticipated well, beat Marc Wilson to the ball and hooked home from a tight angle.
A point was arguably more than Mark Hughes' side deserved for such a listless showing but Hull must surely have feared worse when they were asked to play the majority of the game at a numerical disadvantage.
He shook off Chester to leave himself clean through but was judged to have fouled the defender in the process.
It was the least Hull's battling response deserved and it was the hosts who ended the half pressing for a second as Stoke lost all semblance of calm.
McGregor was called on twice in as Reebok Answer 3
All the momentum lay with the Irishman, who was left in a heap by an outstretched leg that had denied a clear scoring chance.
Chester was not so lucky next time, when Livermore's underhit back pass invited Glenn Whelan to charge at goal.
Andrew Robertson was the chosen outlet and he began to find his range with a couple of teasing crosses from the left.
Referee Jonathan Moss did not hesitate to produce the red card, leaving the hosts with a mountain to climb.
Stoke strike lucky to take a point Express Star
The respective managers quickly re drew the battle lines, Hughes sending Peter Crouch on as a second striker and Steve Bruce replacing Ince with another defender in Liam Rosenior.
Stoke's final substitute, Charlie Adam, was into the game immediately, firing a 25 yarder that McGregor flicked over the crossbar.
The tactical reshuffle was relatively simple, Ahmed Elmohamady and Robertson swapping wing back to full back and Tom Ince dropping deeper into midfield.
Indeed, for much of the 76 minutes they played with a man disadvantage but they were the better side, out passing, out chasing and outworking a Potters team that lacked direction.
Hull's opener just seconds later.
Hull were left to fume with the goal coming from a Stoke throw in which should have gone the home side's way.
It took a wrongly awarded throw in and an unusal bounce of the ball to breach them, Bardsley hitting Bojan's cross into the ground and watching it rear up and bounce off the frame of the goal.
Hughes' half time words seemed to have some effect, though, with his side much improved after the break.
However, Stoke rescued a result, and a first point of the season, when Phil Bardsley's angled shot ricocheted off the inside of the post and against goalkeeper Allan McGregor and was nudged over the line by the waiting Shawcross.
The ball looped invitingly for Diouf but his cushioned header landed a yard wide.
The Tigers had James Chester sent off after only 14 minutes but appeared on course to claim an unlikely win after Nikica Jelavic's instinctive finish gave them the lead shortly before half time.
briefly disorientated, allowed it to come back off him instead of gathering it and Shawcross did just enough his effort confirmed by goal line technology.
many minutes, saving low from Shawcross following a free kick routine then diving in front of Diouf to smother a Saucony Liberty Pack cross.
Hughes responded by sending on summer signing Bojan Krkic four minutes before half time, with Whelan sacrificed, but he was unimpressive in Reebok Answer 14
Stoke had their best chance yet when Bojan's deft pass played in Crouch after 64 minutes but a clumsy touch allowed McShane to make a covering challenge.
The game was gradually building towards Stoke's response but Hull did not seem liable to crack.
McGregor, Reebok Legacy Lifters
Yet despite Hull's positive start, exemplified by the busy Stephen Quinn, Stoke were threatening on the break.
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