Serbian Jankovic has top billing in the inaugural WTA International hard-court event in China and she saw off the challenge of Alla Kudryavtseva in a 6-2 0-6 6-0 win.
Jankovic, who had beaten the Russian on their only other encounter at the 2011 Australian Open, was broken three times and won just four points on her own serve in a sloppy second set.
But she duly delivered a set to love of her own in the decider to set up a second-round clash with Zheng Saisai who defeated Timea Babos 7-5 6-4.
However, seventh seed Puig fell in the first round after a 7-5 6-1 reverse to Andrea Hlavackova.
Puig sits some 111 spots above Hlavackova in the official world rankings and had won both of their previous encounters, but the Czech was in superb form to break serve on five occasions to ensure her progression.
Second seed and home-favourite Peng Shuai faced difficulty against qualifier Shahar Peer but eventually won through 7-5 4-6 6-2.
Varvara Lepchenko, seeded fourth, wasted little time in a dominant 6-1 6-0 victory over Lyudmyla Kichenok, while the latter's twin sister, Nadiia Kichenok, put up a braver fight in a losing 5-7 6-1 6-2 effort against eighth seed Ajla Tomljanovic.
Elsewhere, there were also wins for fifth seed Zhang Shuai, Hsieh Su-wei and Elizaveta Kulichkova.