Boro could hardly have got off to a better start when Stewart Downing, whose goal in the first game was his only strike of the season to that point, beat Robert Green with a perfectly-executed free-kick.
Tuncay dispatched a well-controlled volley into the back of the net for the Teessiders' second.
Gareth Southgate: "Liverpool have beaten Real Madrid tonight, so we have got a fair game on our hands on Saturday.
"But I am just chuffed for the players because it all ends up falling on them and they deserve that result because they have kept driving on in this last few weeks in particular.
"The FA Cup is about dreams and keeping those dreams alive.
"I have played in a few quarter-finals away from home and won. Being away in the quarter-final sometimes isn't a bad thing because you are expected as the home team to win.
"We will be rank outsiders for the game, but that's the beauty of the cup, so we will relish the challenge."