Errani, the only seeded competitor taking to the court on Day 1 of the Madrid event, breezed past Radwanska 6-3, 6-1, rebounding from a first-round exit two weeks ago at Stuttgart.
Elsewhere at the Magic Box Saturday, Slovakia's Daniela Hantuchova downed American Sloane Stephens, 6-3, 7-5 and Belgium's Kirsten Flipkens bested Germany's Mona Barthel, 6-4, 6-1.
Also Saturday, Julia Goerges of Germany bested Serbia's Bojana Jovanovski, 6-4, 6-1, Britain's Laura Robson overcame Slovakia's Magdalena Rybarikova 6-7 (4-7), 6-2, 6-2 and Spain's Lourdes Dominguez Lino defeated Romania's Simona Halep, 6-2, 4-6, 6-4.
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