She led 5-2 and had a match point before Cibulkova came back to make it 5-4 but Erakovic broke again to take the match with a winner down the line.
"Relief, happiness, just very pumped, adrenaline,'' Erakovic told Newstalk ZB while trying to describe her emotions afterwards. "I had some good support in the crowd, which was great. It was just a good feeling.
"I was nervous. Even at 5-2 [in the third set] it was a really tight game. I had a match point there on her serve and after that I was really tight. I felt I played a good game at 5-4 and tried to be aggressive and go for the returns and I think it paid off.''
It is a remarkable turnaround for Erakovic, who had won only one of her seven singles matches going into the tournament since winning her maiden WTA title in February.
It is also one of the biggest wins of her career, perhaps ranking behind only her defeat of then world No 4 Victoria Azarenka at the Bank of the West Classic in 2011. She is also guaranteed to take home $97,000 in prizemoney.
The victory means this is the furthest Erakovic has progressed at Roland Garros after a previous best of reaching the second round in 2008. She becomes the first New Zealand woman to reach the third round at the French Open and equals her best performance at a grand slam after reaching the third round in Wimbledon, also in 2008.
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 semifinals 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.