Strycova, from the Czech Republic, moved on to the second round with a 7-6 (7-2), 6-1 victory over Canadian Marie-Eve Pelletier in which she won 63 percent of her second-serve points and served two aces.
Also advancing was fourth-seeded Wozniak, who battled Russian Elena Vesnina to a 6-2, 4-6, 6-3 win. Wozniak won an efficient 75 percent of her first-serve points.
Seventh-seeded Petra Martic of Croatia ousted American Jessica Pegula 6-4, 7-5.
Americans Melanie Oudin and qualifier Lauren Davis also advanced with Day 2 victories.
Other Tuesday winners included Canadian Eugenie Bouchard, Slovenian Petra Rampre, Georgia's Anna Tatishvili, French qualifier Kristina Mladenovic and Croatia's Mirjana Lucic.