In the tournament's first match at Arthur Ashe Stadium, Radwanska needed only 63 minutes to improve to 28-2 in Grand Slam openers. She won 15 of 21 points at the net.
Errani will next face seventh-seeded Silvia Soler-Espinosa of Spain, who advanced after Croatian opponent Mirjana Lucic-Baroni retired while trailing 6-2, 1-0.
Sania and Mattek-Sands had a great chance of winning the second set after losing the first set tie-break when they led 3-0 but lost the next six games in a reel to bow out of the championships.
Dominika Cibulkova gave Slovakia a 1-0 lead over Spain in their Fed Cup World Group playoff by beating Lourdes Dominguez-Lino in straight sets on Saturday.
Top seed Roberta Vinci was ousted from the $220,000 WTA Monterrey Open on Wednesday, with Russian Nina Bratchikova upsetting the Italian 7-5, 7-5 to reach the quarter-finals.
Fourth-seeded Alexandra Dulgheru of Romania and No. 7-seeded Mathilde Johansson of France were knocked out in the first round of the Monterrery Open on Tuesday.
Maria Sharapova put Russia 1-0 up over Spain in their Fed Cup World Group first round tie here on Saturday by defeating Silvia Soler-Espinosa 6-2, 6-1.