The Finn ended his 17-match losing streak against top-10 players with a 6-4 4-6 7-6 triumph.
"This was one of the matches that I will remember," the 31-year-old told reporters.
"I felt great on the court. The crowd was so into it. Always when I felt tired after a long point, they were almost like standing and clapping. That helped me.
"I really enjoyed that moment on the court. It was unbelievable to play there."
Nieminen will play world number one Novak Djokovic in the last eight.