Venus took the first set while Kvitova won the second set and the deciding set went into a tie break with Venus holding her nerves to edge out her opponent.
After her victory, Venus Williams said: “I have to say I felt every single one of you guys behind me, all 23,000. I didn’t want to let you guys down.
“Everything she has gone through is unbelievable. It’s wonderful to see her back and playing. I was so excited to see her playing well, to be able to play her. I hope we have more matches like that.”
“I would love the top four, five to be that (American) again. That would be huge.
“When situations get hairy and it’s crazy out here, I try to tell myself to enjoy the competition and enjoy the battle.”
Venus Williams is the second of potentially four American women into the semi-finals.