Abdoulaye Faye's deflected goal midway through the second half looked to have reignited the Hammers' npower Championship promotion bid.
James Tomkins could have killed Boro off with a close-range header only to see Jason Steele pull off a stunning save, pushing the ball on to the bar.
But the east Londoners have not won a league match on a Tuesday night in front of their own fans in more than two years, and Nigerian forward Ogbeche's superb strike ensured that sorry run continued.
In fact it could have been worse for the under-fire hosts but Lukas Jutkiewicz hit the bar and Ogbeche had a shot cleared off the line at the death.
West Ham last week