Azarenka battled with her serve early in the first set against the young Belgian, who played valiantly after stepping in for world number two Simona Halep, who pulled out with an Achilles problem.
Azarenka, currently ranked 22 in the world, served seven double faults and lost her opening service game but broke back in a marathon fourth game to take control of the contest as her novice opponent faded.
The Belarusian whizzed through the second set in just over half an hour and sealed victory on her first match point when Bonaventure sent a forehand return long.
Azarenka will next play U.S. Open finalist Roberta Vinci of Italy for a place in the semi-finals.
Nishikori also reached the final of the New York grand slam in 2014 and his game looks suited to the hot and fast conditions likely to await the players in Melbourne when the year's first major gets underway on Jan. 18.
The Japanese second seed needed only 90 minutes to brush past Kukushkin and will await the winner of Wednesday evening's encounter between home hope Bernard Tomic or Czech Radek Stepanek.
(Writing by Patrick Johnston in Singapore; editing by Sudipto Ganguly)
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