Stakhovsky eliminated Youzhny 7-6 (13-11), 6-2 in a match that lasted an hour and 45 minutes at the Khalifa Tennis Complex in Doha.
Fifth seed Viktor Troicki of Serbia advanced with a 6-1, 7-5 decision over Daniel Gimeno-Traver of Spain and No. 7 Ivo Karlovic of Croatia ousted Italian Fabio Fognini 6-4, 6-4.
Other first-round winners included Belgian Olivier Rochus and Evgeny Korolev of Russia.
World No. 1 Roger Federer is the tournament's top seed. Federer will take on Christophe Rochus of Belgium, older brother of Olivier Rochus, in his first match Tuesday.
The tournament is considered a tuneup for the Australian Open, the year's first major, which begins Jan. 18 in Melbourne.