Peng and Hsieh also won the women's doubles title at Wimbledon and the WTA Championships in 2013. Peng became the first Chinese player to hold the top spot in the rankings.
"We are really happy we (are) doing so well this time, especially under the pressure," Hsieh said. "We are No. 1 in the world."
Errani and Vinci were bidding to clinch a fifth Grand Slam title together after winning the women's doubles at Roland Garros and the U.S. Open in 2012, and the Australian Open in 2013 and 2014. They were playing their third straight French Open final.
Peng and Hsieh were put under pressure early. However, the Italian team blew a 3-1 lead in the first set as Errani dropped serve three times.
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The Asian pair won six straight games in the second set, saving three of four break points, and clinched the victory when Errani sent a forehand return wide.
"They know how to do a bit of everything," Vinci said.
"They're very smart on the court."
Peng and Hsieh had a low-key celebration, simply sharing a hug at the end of the match before kissing the trophy. "Cannot like do (something) too dangerous. We are not going to get injured first," Peng said.