Konta recovered to take the second set and level the match but Williams, who won the Rome title in 1999, came back spectacularly with a decisive double break in the third set.
The American will meet either defending champion Garbine Muguruza or Germany's Julia Georges in the quarter-finals.
Estonian qualifier Anett Kontaveit, who defeated world number one Angelique Kerber on Wednesday, continued to build momentum with a 6-1 6-1 victory over 16th-seed Croatian Mirjana Lucic-Baroni.
Sixth-seed Simona Halep marched past Russia's Anastasia Pavlyuchenkova 6-1 4-6 6-0 in the other third-round clash. The Romanian will face Kontaveit in the quarter-finals.
Unseeded Daria Gavrilova stunned Russia's Svetlana Kuznetsova to reach the final eight of the competition with a 2-6 7-5 6-4 victory.
(Reporting by Hardik Vyas in Bengaluru; editing by Richard Lough)
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