The seventh seed, who won the Roland Garros title in 2008, suffered a minor glitch in the second set when she lost eight points in a row as Vekic rebelled to move a break up for 2-1.
But that was as close as the world number 165 came to an upset as Ivanovic won five of the six remaining games on court Suzanne Lenglen.
Ivanovic, who has never made it past the last 16 here since her triumph, next faces either Russian Elena Vesnina or another Russian, ninth seed Ekaterina Makarova.
(Reporting by Julien Pretot; Editing by Sudipto Ganguly)
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