
Their winning goal, which came six minutes from the end of extra time, was the result of a massive slice of fortune as Mido's shot was deflected off the post and off the back of Paddy Kenny's head into the net.
Kenny had been the best player on the pitch until that moment, saving Sheffield United on countless occasions.
Neither side covered themselves in glory, but Middlesbrough had the better of the chances, though United were a bit unlucky not to get a penalty late on.
Middlesbrough will now play Championship side Cardiff City on March 9 in the quarter-finals.