Youzhny, a semi-finalist at Flushing Meadows in 2006, was slow out of the blocks but Robredo failed to capitalise on two early break-point opportunities with the score at 1-2.
And those missed chances proved crucial as the Russian took his only opportunity to break in the 11th game to open up a 6-5 lead before serving out the set.
Robredo engineered an opening when 0-30 up in the second game of the second set but Youzhny recovered to hold serve as the Spaniard failed with a TV review of a forehand on game point.
And Youzhny capitalised on that let-off by claiming a break with an excellent angled backhand in the next game.
Youzhny raced through his next service game and another break quickly followed as Robredo began to display his frustration by banging his racquet on the ground.
But that show of petulance did not help as Youzhny closed the set out.
Robredo recovered his composure in the third set and, after surviving a break point early on, he struck in the 10th game to give himself hope of pulling off an unlikely comeback.
But those hopes were quickly ended as Youzhny broke in the fifth game of the fourth set to take a 3-2 lead and he held his serve to go on and secure the win.
He will now face the winner of the meeting between Sam Querrey and Stanislas Wawrinka which was ongoing last night.