Serena, whose sister Venus was beaten in three sets by Germany's Andrea Petkovic earlier in the day, initially struggled on a fast playing surface but gradually found her range to seal victory in just under two hours.
"It wasn't very easy," Serena, who will face Petkovic in the last four on Saturday, told ESPN. "Ana plays very well. This surface was a little fast so she was hitting a lot of shots.
"But I just fought it and stayed in there. I just tried to do everything that I worked on."
Serena was broken in the first game of the match, then double-faulted to lose serve again in the fifth and trail 1-4 before losing the opening set in just under 30 minutes.
However, the American used her crunching ground strokes to great effect in the next two sets and, helped by 10 aces, she wrapped up the win when her opponent sent a forehand service return wide.
Asked how she had adjusted to the slick conditions on the hardcourt surface, Serena replied: "You just start running. You have to see faster, you have to move faster, you have to turn faster, everything you have to do just a little step faster.
"But I am feeling really good. I had some long rallies and I wasn't out of breath at all so I was like. 'Yeah'," added Serena, who suffered a surprise third round loss at Wimbledon in her previous tournament.
Earlier, the eighth-seeded Petkovic battled past Venus Williams 6-1 3-6 7-5 in a pulsating match lasting nearly two hours.
Former world number one Venus, who had upset fourth seed Victoria Azarenka in straight sets in the previous round, fought back from 6-1 and 2-0 down to level the match but the German regained control in the decisive third set.
"She played really well, and I wasn't as on today as in previous days," said Venus. "I came close, but she was just too good. She's been having a great year and she managed to play those last few games just a little bit better."
In the other two quarter-finals, third-seeded German Angelique Kerber swept aside Spaniard Garbine Muguruza Blanco 6-2 6-1 while Varvara Lepchenko crushed fellow American Sachia Vickery 6-1 6-0.
"It was a very good match for me," said Kerber. "I played good, aggressive tennis. I knew before the match it would be a tough one because she hits the ball very fast, so I was just trying to play point by point. I'm happy about my game today."
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