The British No1 was on course to avenge her Wimbledon semi-final defeat by the American but Williams turned the match to win 5-7 6-1 6-2 and move into the quarter finals.
But Williams, a 10-time Grand Slam singles champion, cruised through the second set to even the score, with Konta requiring a medical timeout before the deciding set.
Meanwhile former world number one Kerber was made to sweat in her gruelling clash with China s Wang Yafan before prevailing 6-7 (1/7), 7-6 (7/5), 6-3.
The second set was another streaky affair, but Williams gained a decisive break for a 5-3 lead and fought off a break point in the next game to force the decider.
JO KONTA will drop out of the world's top 20 after the defence of her Miami title was ended by Venus Williams.
Bertens' confidence grew as Venus sent a forehand long to give her the chance to serve for the set at 6-5 and held to complete the seventh game in a row.
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Fifth seed Karolina Pliskova advanced as Zarina Diyas retired when 6-2 2-1 down and she will take on three-time champion Victoria Azarenka, who beat Agnieszka Radwanska 6-2 6-2, with Elina Svitolina also booking her spot in the quarter-finals with a 7-5 6-4 win over Ash Barty.
"I don't see what 37-years old has to do with it", said Williams. "It didn't look good sometimes but that's why you keep playing to the last point, and why I love this game". She hit just one victor in the second set as the American began to dominate and send the encounter into a decider which began with Konta receiving lengthy treatment for a back problem.
Konta saved set point to turn the first set - which lasted just over an hour - in her favour and take the early advantage. She led by a set and a break before Haddad Maia battled back to level the set.
She next faces ninth-seeded Czech Petra Kvitova, who rallied for a 3-6, 6-2, 6-4 victory over American qualifier Sofia Kenin.
Williams faced more breakpoints in the third although this time the danger was averted, thanks to some powerful serves.
Puerto Rico's Puig followed up her upset of second-seeded Australian Open champion Caroline Wozniacki with a 6-3, 7-5 victory over Greece's Maria Sakkari.