Safina is seeking her first title since climbing to the top of the rankings last month, but was pressured in the first set by the 48th-ranked Martinez Sanchez, who mixed up her game with a number of drop shots and net winners.
The Russian then trailed 3-1 in the second set before turning her game around, winning 11 consecutive games to clinch the set and then the decider.
Safina next plays either Venus Williams or Agnieszka Radwanska, who played later.
In the other side of the draw, Kuznetsova ousted Jankovic 6-1, 7-6 (3).
The eighth-ranked Russian, who won the Porsche GP last week for her first title in almost two years, dictated play with her powerful groundstrokes and hit 27 winners - more than twice as many as her opponent.
In the first set, Kuznetsova fired winners from both sides of the court and attacked Jankovic's second serve. Jankovic was unable to convert two set points at 6-5 in the second, and Kuznetsova closed out the match after 1 hour, 38 minutes.
"I played what she liked, and, of course, she has more power than me,'' said Jankovic, who has dropped to fourth in the rankings after ending last season as No. 1. "Then I started to a little bit change the way ... I was playing. But it was a little bit too late.''
Kuznetsova, the 2004 US Open champion, was a runner-up in Rome two years ago, when she lost to Jankovic in straight sets.
She will next play the sixth-seeded Victoria Azarenka of Belarus, who defeated Kaia Kanepi of Estonia 7-6 (5), 6-3 in a sloppy match that saw 13 breaks of serve - including 10 in the first set - and numerous unforced errors.
"It was a pretty bad match from both of us. I'm just happy I managed to stay calm in the second set and pull it off,'' said the 19-year-old Azarenka, who has won three tournaments this year and jumped to No. 8 in the rankings.
"Definitely I need to step it up and be tough tomorrow.''
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