Azarenka, who lost the U.S. Open final in New York on Saturday and flew in to Rome to beat Venus Williams in a two-hour first-round match on Wednesday, needed only an hour to dispatch the Australian Open champion.
American Kenin, who received a bye into the second round, had no answers to Azarenka's consistency from the baseline and made a string of errors, finishing the match with 22 unforced errors while Azarenka made only four.
Kenin did not create a single opportunity to break the 31-year-old Belarusian while Azarenka won an overwhelming 89% of her first serve points at the Foro Italico while converting six of 11 break points.
"I think consistency was key to the scoreline, I feel like I played really smart tennis today," Azarenka told reporters. "I was able to use a lot of the court and different pace.
"I went for bigger serves when I needed it, I went for variety. I wasn't feeling very good in the warm-up today but it worked out in the match."
Earlier, former French Open champion Garbine Muguruza was tested by American teenager Coco Gauff before the Spaniard advanced to the last 16 with a 7-6(3) 3-6 6-3 victory.
Playing in the main draw of a WTA tournament for the first time in her career, Gauff went toe-to-toe with the twice Grand Slam champion before capitulating in the third set when a string of double faults gave Muguruza the advantage.
"I don't look at her as a different player," Muguruza said. "It doesn't matter how old she is. She's a great competitor and she can play very well."
In the men's field, Italian seventh seed Fabio Fognini was ousted in straight sets by Ugo Humbert, with the Frenchman winning 7-5 7-6(4) in a match that started unusually with six consecutive breaks of serve.
Argentine eighth seed Diego Schwartzman also advanced with a 6-4 7-6(1) victory over John Millman to set up a last-16 clash with Poland's Hubert Hurkacz, who stunned U.S. Open quarter-finalist and ninth seed Andrey Rublev 7-6(6) 3-6 6-2.