Danish world number 15 Wozniacki swept aside Belinda Bencic of Switzerland, while Vinci was similarly dominant against Anna Tatishvili at the WTA International hard-court event
Making her first appearance since reaching the last-16 at Wimbledon, Wozniacki displayed imperious form against world number 62 Bencic in a 6-0 6-0 rout.
Teenager Bencic won just nine of 23 points on her first service as Wozniacki eased through in 45 minutes.
Vinci's straight-sets victory over American world number 112 Tatishvili also proved routine as the Italian second seed won 6-2 6-2 to book a meeting with Alexandra Dulgheru, who recovered from losing the first set to see off fellow qualifier Kateryna Kozlova 3-6 6-0 6-3.
Karolina Pliskova was another big name to avoid an upset in the Turkish capital.
In a rain-affected match, eighth seed Pliskova eventually hit top gear against Turkish wildcard Cagla Buyukakcay - progressing 6-4 3-6 6-0.
Fifth seed Magdalena Rybarikova suffered a surprise exit, however, as she fell at the first hurdle to Croatian qualifier Ana Konjuh despite taking the first set.
While 102 places separated the pair in the world rankings, Konjuh fought back to take their inaugural meeting 3-6 6-3 6-4 and set up a clash with Misaki Doi of Japan.
Doi's opponent Monica Niculescu was forced to withdraw due to injury at 5-3 down in the first set.
There were straight-sets victories for Italian pair Francesca Schiavone and Karin Knapp as they proved too strong for Jovana Jaksic and Ipek Soylu respectively.
Despite double-faulting six times and failing to save any of the three break points she faced, Stefanie Voegele of Switzerland had enough to get past Bulgarian Tsvetana Pironkova 6-4 7-5, while France's Kristina Mladenovic saw off Patricia Mayr-Achleitner 6-4 6-2.
There were mixed fortunes for Israeli pair Julia Glushko and Shahar Peer. The former fell to Czech qualifier Katerina Siniakova, but Peer edged past world number 207 Elizaveta Kulichkova in three sets.
Siniakova will face sixth seed Kurumi Nara, who recorded a first victory over Britain's Johanna Konta at the sixth attempt.