It was a tough one-hour-and-17-minute battle on Rod Laver, but the Belarusian eventually prevailed 7-5 6-1 to continue the defence of her only Grand Slam title.
Azarenka said that she found it difficult to break the Russian down in the first set, but is pleased with the level she was able to reach in the second.
"At the beginning it took a little adjustment, because she has such a different game," said Azarenka. "But there are never going to be easy matches in the quarter-finals.
"I am pleased that I was able to produce some of my best tennis."