Donskoy finished off a 6-3, 3-6, 6-4 victory by cashing in the lone break of the third set. Baghdatis saw only one break point after the first set but that one cost him the match.
Donskoy didn't have a break point to deal with in the first set and saved all three of Baghdatis' break chances on the third.
The other Russians on Tuesday schedule went 0-2, however, as Dudi Sela finished the day's slate with a 6-4, 6-3 win over Aslan Karatsev and Mikhail Kukushkin took out Alex Bogomolov Jr. 6-4, 6-1.
Andrey Golubev, like Kukushkin a qualifier from the former Soviet republic of Kazakhstan, claimed a 6-2, 7-5 decision over Oleksandr Nedovyesov.
Tuesday's other result was an all-Italian pairing with Paolo Lorenzi advancing when, after Lorenzi went up 3-6, 7-5, 4-1, Filippo Volandri retired with an injury.
Notable deaths of 2014 [PHOTOS]