The 22-year-old Vandeweghe posted a beaming photo of herself on Twitter holding the trophy.
"I got the first set and I thought to myself: one set to the good, one more to go," she was quoted as saying on the WTA website. "Try and get the early break now and keep the pressure on her."
Vandeweghe entered the tournament as a qualifier and was playing in just her second final, after losing to Serena Williams at Stanford in 2012. With Madison Keys winning in Eastbourne, this marked the first time that two women from the United States had won titles in the same week since February 2002.
Zheng, a grass specialist who reached the final four at Wimbledon in 2008, was unable to find an answer to the powerful serve of Vandeweghe, who had eight aces and won her first service 27 out of 28 times. She broke Zheng in the third and fifth games of the first set, and again in the third game of the second set. The match was over in 70 minutes.
In the men's final, Becker also dominated with his serve -- at first. He broke Bautista Agut twice in the first set and looked comfortable through most of the second, serving up five aces.
But Bautista Agut took control during the tiebreaker. He began to consistently win rallies, nailing crosscourt shots and moving to the net. The Spaniard broke Becker in the third game of the decider and screamed on winning the final.
Bautista Agut came into the final after a busy Friday when he played the remainder of a quarterfinal match against defending champion Nicolas Mahut, as well as a three-set match against Jurgen Melzer.