Martic, the world number 15, lived up to her senior status in the first set but could not maintain the momentum as Cornet fought back to win 3-6, 6-4, 6-4, sealing the outcome with three aces back-to-back in the final game.
"What a match," Cornet, who is ranked 57th, said as she left the court.
Cornet switched coaches to Martic's former mentor Sandra Zanieswka during the off-season and said she initially had trouble adjusting to her new pre-match instructions to be aggressive.
"It was going out of my comfort zone and in the first set I didn't manage to do it but finally I found my rhythm and was playing much better. Thank God my serve helped me," she said with her only three aces coming when she was down 15-30 in the final game.
Cornet, who also rallied from a set down to win her first-round match against Ysaline Bonaventure, will meet Jessica Pegula in the quarter-finals after the American had a comfortable 6-2, 6-2 second-round win over Tamara Zidansek.
Caroline Garcia became the second seed to fall when the eighth-seed from France was beaten by Canadian Eugenie Bouchard.
Both are former top-five stars with Bouchard, the Wimbledon finalist in 2014, now as low as 262 in the World while Garcia is ranked 46th.
In the quarter-finals, Bouchard will play the winner of the second-round match between third-seed Amanda Anisimova and former top-10 player Daria Kasatkina of Russia.
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