The Tigers opened the scoring when Brady was fouled down in the area. He picked himself up and slotted the spot kick into the corner.
Goalscorer Brady had a shot deflected over and Kevin Nolan fired just wide for the visitors, but Hull were inches away from taking a 2-0 lead when Maiga diverted the ball onto his own crossbar.
West Ham appealed for a penalty early in the second half after the ball appeared to strike Rosenior on the arm, but the referee waved away the appeals.
Danny Graham then wasted an opening, striking over the bar from only four yards out.
The Tigers came even closer to doubling their advantage with 20 minutes to play, and Graham was involved again, laying the ball to Jake Livermore, who hit the post.
West Ham again appealed for a penalty, claiming that Livermore handled, but had to settle for a corner.
Hull were then able to see out the final few minutes.