Kenin's serve was broken early in the first set and the American failed to recover as her Ukrainian opponent claimed the frame 6-3.
Kenin, 21, started the second set stronger by breaking Yastremska's serve for a 2-0 lead, but was soon broken twice herself.
World No. 5 Kenin broke back as the pair battled towards a tiebreak, and it was Yastremska who emerged victorious in the back-and-forth decider for a 6-3, 7-6 (4) win.
Meanwhile, world No. 1 Ashleigh Barty progressed with a 6-3, 6-2 victory over Laura Siegemund of Germany, and No. 3 seed Karolina Pliskova also advanced with ease, beating America's Bernarda Pera 6-3, 6-0.
No. 4 seed Belinda Bencic saw off Veronika Kudermetova 6-4, 4-6, 7-6 (4) after two hours and 41 minutes on court, while Australian Open finalist Garbine Muguruza breezed past Ajla Tomljanovic 6-1, 6-2.