The victory set up a quarter-final showdown with fourth seed Patty Schnyder of Switzerland, who registered a 7-5 6-2 win against Russian qualifier Yaroslava Shvedova.
Mirza tore apart Golovin, whom she had lost to in their two previous meetings, completely dictating play with her huge forehand.
The 20-year-old Indian served efficiently, ate up the Frenchwoman's second serves and won the match when she blitzed a forehand winner crosscourt.
"She said to me, 'can you hit it any harder?'" the world number 35 added with a laugh.
"She's one of my closest friends on tour and it's never easy to play a friend.
"I wanted to dictate as I have had trouble with her in the past because she retrieves so well. But I made sure I was aggressive as soon as I got on court."
Sania's doubles partner Israeli Shahar however failed to progress. The fifth seed was beaten by Belarusian qualifier Olga Govortsova 4-6, 7-5, 6-2.
Govortsova has risen quickly up the rankings, qualifying for Wimbledon and reaching the quarter-finals of Cincinnati last week with her aggressive play.
After starting the year ranked 334th in the world, the 18-year-old began the tournament at number 100 but is now assured of a place in the top 70 after reaching the last eight.
The tall baseliner needed two and half hours to overcome Peer, finally wearing down her Israeli opponent with booming shots into the corners.
"I know her and know what she does good and bad," Govortsova told reporters.
"In the last two sets, I hit more winners. I have more experience and I'm not making as many mistakes as I used to."
Govortsova said that Peer had beaten her thrice as a junior but the Israeli was less consistent this time.
"I'm not playing up to my ability," Peer said. "I'm not playing my game. When I'm running to balls I'm not getting there on time and I'm hitting too short."
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