No. 5 seed Kerber's second-round exit is her worst Wimbledon result since 2013 and follows a first-round exit at the French Open in May.
"I play best when I'm down sometimes," Williams said. "I'm just a fighter and never give up."
Williams, who is seeking an eighth Wimbledon title, has never lost in the second round at the All England Club. She is 23-0 against qualifiers at Grand Slam tournaments.
Juvan was making her Wimbledon debut. She lost in the first round of the French Open in May in her only previous Grand Slam appearance.
Davis, ranked No. 95 in the world, next faces No. 30 Carla Suarez Navarro, who beat Pauline Parmentier 7-6 (2), 7-6 (4). Davis had failed to even get through qualifying and only made the tournament as a lucky loser after some of the wild-card spots weren't used.
"It's almost surreal," said Davis, who started the year ranked outside the top 250.
She said the turning point came when she was up a break at 3-2 in the second set and managed to hold after saving three break points.
"I told myself, 'You're strong, you can do it, you belong here,'" Davis said.
Meanwhile, if the pressure is getting to top-seeded Ashleigh Barty, she is doing a great job of hiding it.
The Australian came into the tournament after winning the French Open and a warm-up grass-court event in Birmingham. Now she has won two in a row at the All England Club to reach the third round and stretch her winning streak to 14 straight.
Barty beat Alison Van Uytvanck 6-1, 6-3, needing only 55 minutes on No. 2 Court to advance. It could have been even quicker, but Barty failed to serve out the match at 5-2 in the second set -- the only time she was broken.
"Pretty sharp right from the start," Barty said. "I was able to implement what I wanted to right away and put the pressure straight back on her."
Barty is playing her first tournament as No. 1 but has never been past the third round at Wimbledon. She will next face Harriet Dart, a British wild-card entry making her second appearance at Wimbledon.
The No. 7-seeded Kvitova was broken in the opening game and trailed 5-3 in the first set. Mladenovic served for that set at 5-4 and held three set points but failed to convert, double-faulting away her initial chance to close it out. Mladenovic later double-faulted twice more to get broken and make it 5-all.
Kvitova, who won titles at the All England Club in 2011 and 2014, wasn't even sure she could compete at the tournament this year until right before it began. The left-hander missed the French Open because of an injury to that forearm.
The Associated Press contributed to this report.