The surprisingly lopsided first-round match lasted just 56 minutes, with the 27-year-old Hanescu converting 4-of-8 break points and surrendered just 10 points on his serve.
With his victory Wednesday, Mathieu broke a string of four straight losses against Youzhny.
He faced a service break in set No. 1 and fought back to force a tie-break in both sets.
The tournament's top two seeds, Gilles Simon and Nikolay Davydenko, will play second-round matches on Thursday against Mischa Zverev and Michael Berrer, respectively.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut