The Baggies had been on course to secure a share of the spoils at the Riverside Stadium, but a last-minute goal from Daniel Ayala earned all three points for the home side.
The decisive strike came from a set piece, and Moore has admitted that he had been left disappointed with his side's attempt to deal with a ball sent into the penalty area.
Moore told BBC WM: "It's a sickener to lose it in the final few minutes of the game. We just lost the flight of the ball in the air - I think if we contest it we come away with a clean sheet.
"You've got to give Middlesbrough credit, they changed their shape to combat us. They went with a defensive display and they'll probably feel they got it right in terms of nullifying us.
"It's a disappointing one to take because I thought during the 90 minutes we played the game right."
West Brom remain in seventh place in the Championship standings.