The Italian came through 7-5 7-6(3) after two hours and 16 minutes of tennis.
The first set was a close affair, with both players breaking twice in the opening four games. However, Errani made a crucial break at 5-5 before serving out the set to love.
Cornet struggled to deal with the world number six in the second set as Errani broke serve twice to move 4-0 up.
The French player regained her form, saving match point at 5-2 before breaking twice to move 6-5 ahead. Errani then held serve to take the set to a tie-break, which she won 7-3.