Boro proved their loftier position early on with Marvin Emnes going close as the visitors began to turn the screw.
It was, however, the home side who were the first to strike after 36 minutes. Clive Platt headed down a long ball to the onrushing McSheffrey, who dispatched from six yards out.
Things went from bad to worse for Tony Mowbray's men as, three minutes later, Kevin Thomson was sent off for a second bookable.
The Championship's bottom side doubled their lead 12 minutes after the break through Nimely who headed towards goal. Neither side was sure if the ball had gone over but referee Gary Sutton gave the goal.
Bates then turned the ball into his own net before McDonald reduced the deficit by finishing Emnes's cutback.
The Sky Blues remain bottom despite the win, while Boro stay fourth, eight points adrift of new leaders West Ham.