Ranked 145 on the WTA circuit, Sania scored a 6-4, 6-2 win over Kuryanovich, ranked 61 places below the Indian, in the opening round of the women's singles qualifiers.
Sania broke her rival, playing her first Australian Open, twice in the closely fought first set and also dropped her serve once.
The Indian continued to struggle with her serve and faced four breakpoints in the second set but managed to save three of them.
The Belarusian was equally erratic and had to contend with seven breakpoints out which Sania converted three and sealed the win in one hour and nine minutes.
Sania will next face Russia's Vitalia Diatchenko. Due to injuries and slump in form, Sania's ranking suffered in 2010 and she was forced to go through the qualifiers.
Her one of the best performances in Grand Slams came at Australian Open when she reached the third round in 2005 and 2008.
Last year Sania had made a first round exit here.