Grant swept home in the 69th minute for the only goal to reward a dominant performance from Valerien Ismael’s side as Hull suffered a seventh game without an away goal.
Hull substitute Ryan Longman wasted two late chances and the second-from-bottom visitors have now gone seven games without an away goal, leaving them three points from safety.
But head coach Grant McCann said: “I said to them if we play like that for the rest of the season then we won’t have any problems and we will climb the table.
“We lost a game through a moment of madness. One mistake when we lost a tackle left us split wide open against a team expected to win the division and they punished us.
“The lads are hurting in the changing room because apart from a mad 30 seconds, we went toe to toe with them.
“The team showed what we are about. I’m proud of the team despite that mistake - I could not ask any more of the boys.
“They were defensively solid and the structure of the team was solid.”
Albion created three early chances. Matt Phillips hit the post from Callum Robinson’s cross then Grant’s follow-up from even closer range was turned behind by goalkeeper Matt Ingram.
Robinson was then presented with a golden chance but he delayed his shot then saw two attempts blocked after Taylor Gardner-Hickman’s intelligent through ball.
The visitors had a rare opportunity just after the midpoint of the first half, but Jacob Greaves could only glance wide from a corner.
Albion turned up the tempo soon after the restart to create another opening for Robinson.
Gardner-Hickman beat two players on the right before squaring to Robert Snodgrass, who sprayed the ball wide to Reach for a cross stabbed wide by the former Sheffield United forward.
The catalyst for the victory could be traced back to the 63rd minute when Ismael introduced Jordan Hugill for Phillips.
West Brom forced Ingram into two important saves before they finally broke the deadlock.
Robinson decided to shoot early and forced the Hull goalkeeper to spring to his right to palm the ball away.
Grant then produced a fierce, rising drive that was tipped over the bar.
But Grant swept home from eight yards after a poor pass across his own box by Greg Docherty was intercepted by Jake Livermore, who fed Robinson to square for what proved to be the winner.
West Brom goalkeeper Sam Johnstone saved the blushes of team-mate Cedric Kipre when he scooped the ball behind after the defender diverted George Honeyman’s cross goalwards.
Grant should have made it 2-0 when he ran half the length of the pitch, only to drag his shot wide.
Hull launched a late rally and Ryan Longman had a shot saved by Johnstone before seeing another effort blocked.