The Australian dropped a tight first set to the Israeli qualifier, losing it 7-5. Both women has two opportunities to break, but it was Glushko who took full advantage.
However, the 23-year-old - born in Donetsk, Ukraine - tired from there on in, as Stosur quickly got back on level terms after taking the second set 6-2.
The tide had now turned, with the 2011 US Open champion turning the screw on her opponent.
Stosur completed a 6-3 victory in the third set after two hours and 10 minutes on court. She will now face Carla Suarez-Navarro in the next round.