Begovic was called into action again to thwart Yannick Sagbo's low effort.
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The second half was five minutes old when Glenn Whelan smashed a volley over the bar from 30 yards.
Jake Livermore then hit a volley wide for the home side, but the game was still short of quality.
Stoke went close shortly after as Ryan Shawcross poked wide after the ball had ran through to him from a corner.
Begovic somehow kept out Elmohamady's header at the back post before Arnautovic took too long to get his pass away and Hull cleared.
Mark Hughes's men thought they had taken the lead with 20 minutes remaining, but Ireland's effort was ruled out for offside.
Shawcross then cracked his header against the crossbar as the visitors pushed for the opening goal.