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Victory would have kept second-placed West Brom's destiny in their own hands but they fell behind after four minutes when Chris Willock stabbed home from close range.
West Brom equalised shortly before halftime with a Dara O'Shea header at the back post.
They had chances after the break but were stunned late on when the hosts counter attacked and Emile Smith Rowe, on loan from Arsenal, swept in an angled shot.
Victory will likely spare Huddersfield a second successive relegation but West Brom are now in danger of missing out on a lucrative promotion to the Premier League.
West Brom have 82 points from 45 games but will be replaced in the top two if third-placed Brentford, who have 81 from 44 and a better goal difference, avoid defeat at Stoke City on Saturday. Leeds are top with 87 points and have two games left.