The Czech Republic's Zakopalova made short work of countrywoman Kristyna Pliskova 7-6 (7-2), 6-2 indoors at the Spodek arena.
Britain's Robson, meanwhile, fell victim to Spaniard Lourdes Dominguez Lino by a 5-7, 7-6 (9-7), 6-1 verdict.
In other Day 1 action, Czech wild card Karolina Pliskova bested Spaniard Maria-Teresa Torro-Flor 6-1, 3-6, 6-0 and Germany's Annika Beck dispatched Polish wild card Marta Domachowska 6-3, 7-5.
Also Monday, Croatia's Petra Martic got past Ukrainian Elina Svitolina 6-7 (2-7), 6-3, 6-2.
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