Kenin, who sits at world No.143 three years after her Melbourne Park triumph, was too strong in a 6-2 6-2 first-round win over her 85-ranked opponent on Tuesday.
“I feel like I played well out there. It was tough conditions,” Kenin, who won the Hobart title in 2019, said.
“I’m happy with the way I’m playing, so I’m just going to try and keep it going.”
The 24-year-old American, who has battled injury and poor form since her 2020 Australian Open title, made the second round in Auckland last week, losing to eventual winner Coco Gauff.
Cocciaretto will next meet countrywoman Jasmine Paolini for a place in the quarter-finals.
It comes after fourth seed Sloane Stephens was eliminated in the first round on Monday.