The 31-year-old Slovakian was bothered by a troubling wrist injury in 2014 but also lost in the first round at 12 of her 24 tournaments, including the last four of the year.
Hantuchova, who has linked with Carlos Rodriguez, the former coach of now retired Australian Open champion Li Na, did not make it past the third round at any of the grand slam events last year and finished ranked 62nd in the world.
Monday's victory over an error-prone Errani, the world number 14, was her first win over a player in the top-100 since last July and Hantuchova told reporters that was she was 'nicely surprised' with her win.
Errani was not the only seed to fall on the opening day of the tournament with eighth seed Mona Barthel hammered 6-2 6-1 by Croatia's Ana Konjuh, while 2010 champion Yanina Wickmayer of Belgium was beaten by 18-year-old American Taylor Townshend.
Sixth seed Sloane Stephens also advanced with a 6-3 6-1 victory over Silvia Soler-Espinosa.
Top seed Caroline Wozniacki and last year's beaten finallist Venus Williams will play their first round matches on Tuesday.
(Reporting by Greg Stutchbury; Editing by Sudipto Ganguly)
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