The 21-year-old came through 6-2, 6-3, winning only China's second-ever tennis gold after Li Ting and Sun Tiantian's women's doubles triumph at Athens in 2004.
Zheng's victory came on the same Court Philippe Chatrier at Roland Garros where in 2011 celebrated compatriot Li Na became China's first Grand Slam champion in a landmark moment for the sport.
"Nothing can describe my emotion, every round was super difficult and I did everything I could to get a medal for my country," said Zheng, who has improved on the Olympic record of Li, whose best performance was a fourth place in Beijing in 2008.
"I feel my country will be proud of me, I'm proud of myself. My family are at home, I'm sure they are screaming at the TV.
"I just fight every match. I have a special energy playing for my country. I never give up."
World number seven and Australian Open runner-up Zheng took advantage of settling into a groove faster than her opponent and broke for a 2-0 lead.
Vekic and Zheng went head-to-head with some brutal hitting but the Chinese star's 12 winners to the Croat's three proved crucial and her accuracy shone again with a set-clinching break in the eighth game.
Zheng had saved a match point in her gruelling third-round win over Emma Navarro and then ended four-time French Open champion Iga Swiatek's 25-match winning run at Roland Garros to reach the gold medal match.
Match-hardened by those tests, Zheng held to love in the opening game of the second and broke again for a 2-0 lead as the errors piled up for 28-year-old Vekic.
However, the Croat, who knocked out world number two Coco Gauff in the third round, and saved a match point in her quarter-final win over Marta Kostyuk, hit back with her first service break of the contest.
Zheng thwarted any hint of a revival by converting a fourth break point for a 5-3 lead before sealing victory in the next game.
"I'm disappointed," said Vekic. "But once everything settles I will be proud of myself."
Swiatek claimed the bronze medal on Friday by seeing off Anna Karolina Schmiedlova.
It was a rollercoaster tournament for Zheng, who needed back-to-back three-hour matches to defeat Navarro and then former world number one Angelique Kerber.
In both matches she had to claw back deficits while against Navarro she even found her reputation under attack when the American described her as "cut-throat" with "no respect".
