Abdoulaye Faye's deflected goal midway through the second half looked to have reignited the Hammers' Championship promotion bid.
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.
Manager Sam Allardyce admits his players currently feel anxious every time they play on their own turf, having been booed off following three straight draws.
And that was certainly apparent in a tense first half against a Middlesbrough side hardly bursting with confidence themselves on the back of successive defeats.
Faye headed wide from a corner, Matt Taylor's shot was blocked by Seb Hines, Nicky Maynard's long-range effort was held by Jason Steele and Noble drove over the bar, but there was little to lift the home faithful.
Boro were not faring any better, though, with Adam Hamill firing wide and Jutkiewicz putting a couple of headers too high.
As half-time approached West Ham felt they should have had a penalty for handball but James Tomkins' header seemed to come off the shoulder of Matthew Bates.
Carlton Cole then curled a shot narrowly wide and George McCartney's 30-yard effort was dealt with by Steele, but the hosts trooped off at the break to familiar murmurs of discontent.
But the Hammers went on the attack in the second half and Maynard's chip dropped on to the roof of Steele's net before Cole just failed to get on the end of Taylor's looping cross.
And the slice of luck they needed finally arrived in the 67th minute, courtesy of Faye.
The giant defender rose to meet a long, high ball from McCartney and his header spun off the back of marker Joe Bennett's head and looped over Steele into the far corner.
Tomkins could have killed Boro off with a close-range header only to see Steele pull off a stunning save, pushing the ball on to the bar.
And instead Ogbeche pegged West Ham back seven minutes before full-time, skipping past Danny Collins on the edge of the area and crashing a superb shot past Rob Green.
Suddenly West Ham were clinging on as Jutkiewicz's shot was deflected on to the bar and McCartney cleared off the line from Ogbeche.
But holding out for a draw did not stop more boos ringing out at the final whistle.