Shevchenko overcame a 10-3 disadvantage in aces by converting two of three break points, to 0-for-3 for Khachanov.
Shevchenko advanced to the semifinals Friday against wild card Pierre-Hugues Herbert, who outlasted fellow Frenchman Luca Van Assche 4-6, 6-3, 7-5 on Thursday.
Sofia OpenSecond seed Adrian Mannarino and third seed Jan-Lennard Struff avoided upsets with three-set victories Thursday in the quarterfinals in Bulgaria.
Mannarino of France eliminated seventh seed Sebastian Ofner of Austria 6-7 (7), 6-4, 6-3.
Mannarino advanced to the semifinals Friday against Russia's Pavel Kotov, who defeated eighth seed Marton Fucsovics of Hungary 6-3, 7-6 (3).
Persons:
Russia's Alexander Shevchenko, Canada's Felix Auger Aliassime, Pierre Albouy, Alexander Shevchenko, Karen Khachanov, Shevchenko, Pierre, Hugues Herbert, Frenchman Luca Van Assche, Ugo Humbert, Frenchman Harold Mayot, Lorenzo Sonego, Fabio Fognini, Adrian Mannarino, Lennard Struff, France, Sebastian Ofner, Germany's Struff, Hungary's Fabian Marozsan, Struff, Great, Jack Draper, Lorenzo Musetti, Cem Ilkel, Mannarino, Russia's Pavel Kotov, Marton
Organizations:
Swiss, Basel, Canada's Felix Auger Aliassime REUTERS, Khachanov, Sofia, Thomson
Locations:
Jakobshalle, Basel, Switzerland, Metz, France, Italy, Bulgaria, Mannarino, Austria, Turkey, Marton Fucsovics, Hungary