"Out of the previous nine times we've played I think we've gone three sets at least six times so I know she plays really well," the top-seeded American said in an on-court interview.
"She's been number one, she's won a grand slam. I knew it was going to be a big test for me today and I'm really happy I was able to get through."
Ivanovic did well to contain her opponent's powerful serve, most notably in the first set when she broke three times in four games to pull ahead 5-3 before delivering an incredible backhand winner to hold to love.
A topsy-turvy second set saw Williams grab the first three games and Ivanovic the next four before the world number one roared back to win the next three.
The Serb then grabbed the opening two games of the final set but Williams shifted into a higher gear to win six in a row and advance.
The American landed only 31 percent of her first serves in the opening set and 47 percent in the second but was much more efficient in the third, chalking up 68 percent.
"I just decided that if I wanted to stay out here and keep playing I had to pick my game up a level," said Williams who now has a 9-1 career record against Ivanovic.
"I had to do a 'Serena'. I kind of like that actually."
Up next for Williams will be 14th-seeded Ukrainian Elina Svitolina, a 6-4 2-6 6-0 winner over Czech Lucie Safarova, seeded seven.
In Friday's other quarter-finals, third-seeded Romanian Simona Halep dispatched Russian Anastasia Pavlyuchenkova 7-6 6-2.
She will now face a semi-final against Serbian Jelena Jankovic who beat Slovakian qualifier Anna Karolina Schmiedlova 6-4 6-2.
Halep and Jankovic last squared off in the BNP Paribas Open final in March when Halep prevailed in a three-set battle.
(Reporting by Frank Pingue; Editing by Tony Jimenez / Ian Ransom)
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