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The Kiwi No. 1 beat Cibulkova in three sets 6-2, 2-6, 6-4 in a match that lasted two hours and 38 minutes.
Erakovic went into the second round match having lost the previous two meetings with the Slovakian but held her nerve in the final set to hand the 2012 quarter-finalist an early exit.
With Erakovic leading 5-2 in the third set, Cibulkova came back to make it 5-4 before the New Zealander took out victory with a break at match point in the 10th game.
Erakovic had more unforced errors than Cibulkova in the match (38-31) but finished well ahead on winners (38-17).
The victory means this is the furthest Erakovic has progressed in Paris after a previous best of reaching the second round in 2008. She becomes the first New Zealander to reach the third round at the French Open. It equals her best performance at a Grand Slam after reaching the third round in Wimbledon, also in 2008.
She will now meet American 17th seed Sloane Stephens in the next round after Stephens defeated compatriot Vania King 6-1, 6-3.
Erakovic and Stephens have meet just once previously with Erakovic coming out on top at the US National Indoor Tennis Championships in Memphis last year in three sets. They have never met on clay.
Stephens reached the semi-finals at the Australian Open earlier in the year after upsetting fellow American Serena Williams in the quarter-finals.
Erakovic was set to open her doubles campaign today with partner Cara Black against German duo Angelique Kerber and Andrea Petkovic but their first round match was moved to tomorrow due to a rain delay.