The 19-year-old wildcard, who has been struggling with an ankle injury since the beginning of the season, needed just one hour and 20 minutes to beat Czech Safarova and set up a clash with world number one
Watson rattled off the first three games of the match, including a break in the second game, before breaking for a second time to wrap up the first set.
Safarova bounced back by breaking in the opening game of the second set but Watson immediately regained her composure and broke back before storming through the rest of the match.
World number one Azarenka, who thumped Watson 6-1 6-0 in the opening round at the Australian Open, stretched her unbeaten run to 24 matches this year with another easy win in her first appearance in Miami.
Her latest victim was
Azarenka is off to the hottest start to a season since Martina Hingis won her first 37 matches in 1997. Her wins this year include the Australian Open and Indian Wells last week, but she realizes she now has a target on her back.
"I felt that she had nothing to lose, so she went for her shots, she was really aggressive," Azarenka said.
"I think I could have been more aggressive and play more my game, but I found the right shot selection and the right opportunities when I needed."
Former world number one Williams, playing in her first WTA event since last year's U.S. Open, broke Czech Kvitova three times in the final set.
"The third set was very cruel for me," left-hander Kvitova said. "The first two games I had game points and break points, but I didn't make it.
"Then it was tough to still focus on my plan for the rest of the match. I knew what I had to do today but, if I wasn't returning very well, it's tough to play against her.
"She looks very ready, very fit. She's back," the Czech said of Williams, who revealed at the US Open she had been diagnosed with Sjogren's syndrome, an auto-immune disease that causes fatigue and joint soreness.
Elsewhere, ninth seed