The No.1-seeded Jankovic looked headed for a straight set victory after winning the first set and twice going up a break in the second set - including serving for the match leading 64 54 - but the tenacious Cohen pushed it to a third set before falling in a three-hour, 18-minute marathon, 64 57 62.
"It was really long for a first round," Jankovic said after the match. "She was playing these high balls, which were difficult for me to attack. But I was able to stay strong mentally and physically, and I told myself to keep being aggressive and going for my shots. I kept doing that until I won the match."
Jankovic is no stranger to clay - five of her 12 career WTA titles have come on the slow surface, specifically Budapest in 2004, Charleston and Rome in 2007, Rome in 2008 and Marbella in 2009.
"I know her," Jankovic said of her next opponent. "She's a very nice girl and a very good player. I practiced with her a long time ago. It will be a difficult match - I'll just go out there tomorrow and try to play my best tennis, and we'll see how it goes. Every match is tough these days, really.
"I've had a lot of ups and downs the last few years and some things haven't gone my way, but I'm really working hard and trying to play hard. I'm hoping everything improves from now on."
The first three second round matches also took place, with a big milestone among the results: Teliana Pereira battled for more than three hours to pick off the tournament's No.2 seed, Alizé Cornet, and become Brazil's first WTA quarterfinalist since this very tournament all the way back in 1999.
"Woohoo!" Pereira exclaimed upon being told the news - and among the crowd around her was former Top 30 player Alexandra Dulgheru, who said they should put a statue of Pereira up in her hometown.
Pereira - who, at No.156 in the world, is Brazil's only player in the Top 300 - is the first Brazilian to reach the quarterfinals of a WTA tournament since Vanessa Menga achieved the feat here in 1999.