Phau beat the Russian 4-6, 7-5, 6-1 and will compete in his first quarterfinal since Stuttgart in 2012.
Another qualifier and a lucky loser also made the last eight.
Youzhny, the 2012 champion, was playing well until he lost his serve in the second set when he foot-faulted. Phau used the opportunity to win four games in a row and turn the match around.
''It was not the foot fault that caused me to lose the match,'' Youzhny said, adding he felt sick during play.
Dudi Sela of Israel also advanced, defeating Lukas Lacko of Slovakia 6-3, 6-2.