The Baggies' first win on Teesside since 2009 was secured eight minutes from time when Boro looked on course to claim a vital point in their own bid to climb away from relegation trouble.
Both teams had their chances in the two halves but there had been no way through as the Baggies found it hard against a Middlesbrough team threatened with falling into the drop zone.
Despite winning just twice in 12 matches under head coach Jonathan Woodgate, Middlesbrough caused problems.
Albion, who had lost just once this season, had to make one change to their team because of a knock sustained to Matt Phillips.
There was another fine save from Assombalonga on the stroke of half-time. George Saville headed the rebound towards goal and just as it seemed it may drop under the bar Kyle Bartley was there to nod over.
Pereira tried an acrobatic kick from Diangana's high pass early in the second half that was gathered by Randolph, and that was followed by Marvin Johnson shaving the side-netting at the other end.
Johnson sparked into life. His run led a counter-attack that ended with Johnstone needing to be alert to block a drive from Howson, while the West Brom goalkeeper made a stop with his feet just seconds later from Saville's low effort.
Johnson also needed to be alert at the other end when he made a fantastic sliding tackle to prevent substitute Kyle Edwards from getting a shot at goal in the penalty area after he was played in behind.
When West Brom pushed for a winner, Boro almost took the lead against the run of play. Ashley Fletcher side footed an effort over the bar from inside six yards after Johnstone had got down low to turn away Saville's effort away.
Randolph did his best to stop Edwards' effort from Diangana's run and cross, but it was not enough as Welsh international Kanu was on hand to bundle a finish over the line to claim the points.