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|Venus Williams had to dig deep to battle past Kiki Bertens in Miami. (Photo: AFP/Al Bello)|
Williams, ranked eighth in the world, labored nearly three hours to beat 29th-ranked Bertens 5-7, 6-3, 7-5.
The 37-year-old American was cruising with a 5-0 lead in the opening set, but she squandered three set points as Bertens reeled off seven straight games to snatch it.
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.
The seven-time Grand Slam champion would have to dig deep when she fell 1-4 down to her Dutch opponent in the third.
Bertens served for the match at 5-3 but was unable to convert two match points as Williams broke to put the match back on serve.
Williams would face one more match point against her own serve in the next game, but she held on as Bertens appeared to struggle with cramps in the final games.
Williams sealed the win with an emphatic forehand winner, giving a little leap of elation.
"It didn't look good sometimes, that's why you keep playing until the last point," said Williams, who lifted the trophy in Miami in 1998, 1999 and 2001. "And that's why I love this game."
Konta, whose victory here last year propelled her into the top 10 in the world rankings, advanced with relative ease, beating Belgian Elise Mertens 6-2, 6-1.
Konta exchanged early breaks with Mertens, but gained the upper hand with a break for 3-2 when the Belgian double faulted four times in one game.
It launched a streak of four straight games for Konta that gave the Briton the opening set.
She powered through the second to reach the fourth round, where the 26-year-old will try to get past the last 16 of a tournament for the first time this season.
Fourth-seeded Elina Svitolina and sixth-seeded Jelena Ostapenko, the reigning French Open champion, also advanced, as did 2016 Olympic gold medalist Monica Puig.
Ukraine's Svitolina beat Daria Gavrilova 4-6, 6-0, 6-1, her fourth straight win over the Australian.
In a match of tense rallies Svitolina finished with 28 winners and won 70 per cent of her first-serve points.
Svitolina will battle for a quarter-final berth against another Australian, 21st-seeded Ashleigh Barty, who beat Croatia's Petra Martic 6-4, 6-3.
Latvian Ostapenko defeated Brazilian Beatriz Haddad Maia 6-2, 7-6 (7/2). She led by a set and a break before Haddad Maia battled back to level the set.
Ostapenko twice served to avoid a third set. She fell behind 0-2 in the tiebreaker before winning seven straight points for the victory.
"In the tie-break, I showed my great tennis, and I won the match," Ostapenko said. "I'm really glad it happened that way."
She next faces ninth-seeded Czech Petra Kvitova, who rallied for a 3-6, 6-2, 6-4 victory over American qualifier Sofia Kenin.
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.