Begu, who is ranked 10 spots behind the 63rd-ranked Hercog, saved three set points in the tiebreaker.
Earlier on the red clay courts at the Country Time Club, seventh-seeded Alize Cornet of France beat Mariana Duque-Marino of Colombia, 6-3, 6-2; Alexandra Cadantu of Romania eliminated Kiki Bertens of the Netherlands 6-3, 6-2; and Lourdes Dominguez Lino of Spain defeated Yulia Putintseva of Kazakhstan 6-2, 6-4.
Wimbledon runner-up Agnieszka Radwanska withdrew from the tournament citing respiratory problems, saying that the humid weather in London affected her in Saturday's final loss to Serena Williams.
French Open runner-up Sara Errani is the tournament's top-seeded player.