"It was one of the best matches I've ever played," said Thiem, who also reached the last eight at Kitzbuehel in July.
Haas reached his 12th quarterfinal of the season by defeating Miloslav Mecir of Slovakia 7-5, 7-6 (8).
Playing in his first ATP event, Mecir missed three set points on the German's serve at 5-4 in the opening set and two in the tiebreaker of the second. The 240th-ranked Slovak, the son of 1988 Olympic champion Miloslav Mecir, came through three qualifying rounds and stunned Spain's Pablo Andujar in the first round.
"I had not seen him before but I was told he plays like his father," said Haas, who won the event in 2001 and next plays fifth-seeded Radek Stepanek. "It was a tough match. I had to get used to the slow court and the slow balls here. I have to be satisfied as I made the important points."
The 176th-ranked Bemelmans reached the last eight for the first time in his career. He will face eighth-seeded Lukas Rosol of the Czech Republic. Kohlschreiber missed out on his first quarterfinal since reaching the final in Stuttgart in July.
Also, Robin Haase of the Netherlands missed a second-set match point against seventh-seeded Vasek Pospisil of Canada before closing out a 7-6 (3), 6-7 (5), 6-2 win.
Haase, who earned both of his career titles in Austria, lined up Fognini, who defeated Lukasz Kubot of Poland 7-5, 6-2. The Italian failed to serve out the opening set at 5-4 but won eight of the next 10 games to reach his 10th quarterfinal of the year but only his second on hard court.