Serbia's Jankovic, a former world No. 1, had to go to three sets but managed to overcome American Julia Cohen 6-4, 5-7, 6-2 at Centro de Alto Rendimiento in Bogota.
She advanced to the second round, where she'll take on 2010 champ Mariana Duque Marino of Colombia, who got past Spaniard Estrella Cabeza Candela 6-3, 3-6, 6-4 in another opening-round contest.
France's Cornet, meanwhile, fell short of the quarterfinals, tumbling to Brazilian qualifier Teliana Pereira 7-5, 6-7 (2-7), 6-2.
Other second-round matches saw Mandy Minella of Luxembourg dispatch No. 8 Timea Babos of Hungary 6-1, 6-4, while Spain's Maria-Teresa Torro-Flor got past France's Mathilde Johansson 7-6 (7-3), 6-3.