Youzhny made a strong start in the third-round clash, forcing a timely break early on to claim the opening set 6-3 in 38 minutes.
Haas struggled to find consistency on serve and soon found himself two sets down to the 21st seed, with the trainer called for treatment shortly before dropping the second set 6-2.
With victory in sight, Youzhny took his foot off the pedal as unforced errors crept in, gifting Haas a lifeline he grabbed to claim the third set and make the match a genuine contest once more.
The Russian then swiftly regained his composure in the fourth set, breaking twice in quick succession to build an advantage.
Haas showed no signs of giving in and quickly got one break back, but the 35-year-old's struggles on serve continued as Youhzny found some composure under pressure.
Youzhny kept his composure to close out a 6-3 6-2 2-6 6-3 win and advance to the last 16 at Flushing Meadows, where he will face another veteran in Australian favourite Lleyton Hewitt.