World number one Swiatek is yet to drop a set and is favourite for a third Foro Italico crown before she begins her French Open title defence later this month.
The three-time Roland Garros champion arrived in the Italian capital off the back of a thrilling triumph in Madrid, her third WTA 1000 title of the season after also winning at Indian Wells and Doha.
But the 22-year-old didn't have it all her own way against former world number one Kerber, a three-time former Grand Slam champion who at 331st was the lowest-ranked player ever to reach the last 16 in this tournament.
Swiatek battled to the first set in just under an hour and was frequently put to the test by 36-year-old German Kerber, who fell behind after saving five set points in game 10 and then succumbing two games later.
Kerber went two ahead in the second set but Swiatek fought back and after the Pole broke serve in game eight she held steady to close out the match.
Swiatek will face Keys whose 6-2, 6-1 win was suspended for half an hour by a court invasion from activists from Ultima Generazione (Last Generation).
Wearing orange vests, protestors threw a liquid and confetti onto the Pietrangeli court, where Keys was playing, and court 12 where a men's doubles match was taking place.
There were also protestors in the stands with a tournament spokesman telling AFP that at least one person had tried to glue their feet to the floor in order to slow their removal.
"Obviously it's not the greatest feeling when you're on court, your first reaction is kind of your own safety," said Keys.
"I think maybe banning cementing glue from bags would be a start.
"It's obviously something that's becoming more of an occurence and something that tournaments are going to have to figure out how to stop."
Medvedev resumes defence
Daniil Medvedev will continue his title defence against Serbian qualifier Hamad Medjedovic with the men's tournament wide open following a series of big-name exits, the latest being fourth seed Andrey Rublev.
Rome is missing the world's top three men following Novak Djokovic's early elimination on Saturday, with Italian world number two Jannik Sinner and Carlos Alcaraz both withdrawing ahead of the event.
And Russian Rublev, fresh from claiming the Madrid title, suffered a shock third-round defeat by French qualifier Alexandre Muller, 3-6, 6-3, 6-2.
Sixth seed and reigning Monte Carlo champion Stefanos Tsitsipas is still in the fray after beating Briton Cameron Norrie in straight sets 6-2, 7-6 (7/1).
Naomi Osaka's return to the Rome Open was ended at the last 16 by China's Zheng Qinwen who breezed past her four-time Grand Slam-winning opponent 6-2, 6-4 to set up a last-eight clash with third seed Coco Gauff.
Japan's Osaka, who has never won a title on clay, was playing at the Foro Italico for the first time in three years after taking a break from the tour to have her first child and is ranked at 173 in the world.
"I think I'm happy with my performance in Rome. It's been a while since I played well on - I guess I've never played well on clay - so I guess I'm happy," Osaka told reporters.
Later Australian Open champion Aryna Sabalenka will face Elina Svitolina for the first time since their handshake row at last year's French Open.
Belarusian second seed Sabalenka was accused by Ukraine's Svitolina of inflaming tensions surrounding the Russia-Ukraine war after the Roland Garros quarter-final between the pair in 2023.
Sabalenka waited at the net for a handshake after beating Svitolina, even though Svitolina had already insisted she was not going to offer one, as all Ukraine players refused to shake hands with Russian and Belarusian rivals in protest at the ongoing war.
