Halep looked headed for a straight set victory leading 62 43 with double break point for 62 53, but Safarova dug that game out, broke Halep and held one more time to sneak out the second set, 6-4.
After the two traded holds early in the third, Safarova cracked the match open by breaking for 3-2 - and, after getting broken right back, she broke again for 4-3, and held a point for a 5-3 lead. That's when Halep took it to another level, though - the No.6-seeded Romanian reeled off nine points in a row to go from a point for 3-5 to 5-4, 30-0. She eventually closed out the No.26-seeded Czech, 62 46 64.
Halep's extra gear in the third set is just the kind of thing that has helped her burst into the upper echelon of women's tennis over the last 10 months. While she may have lost some of those close ones beforehand, she's now cool, calm and collected - and she's able to let her obvious talents shine.
"I think my game is complete," Halep said earlier in the week. "I can play well with both sides. I don't serve really well, but I keep working on that. And I move well, so I stay close to the baseline.
"I think I'm playing beautiful tennis. Maybe it's because I'm very relaxed on the court now, and that's the secret to winning a lot of tournaments. I just want to keep going like this - it's the right way."
Earlier this year Halep broke the Top 10 for the first time, becoming the third Romanian to break that elite after Virginia Ruzici and Irina Spirlea. With her recent rise to No.7 in the world she passed former World No.8 Ruzici and tied former World No.7 Spirlea for highest-ranked Romanian in WTA history.
But with those two coming from very different tennis generations, does she have any ties with them?
"My manager is actually Virginia Ruzici," Halep said. "She tells me about her experiences all the time and helps me deal with pressure. There's going to be more pressure on me now, so it's good to speak with someone who knows about it. She's not here, but I speak with her every day - she'll be in Miami."
Awaiting Halep in the fourth round is No.18 seed Eugenie Bouchard, who upset No.9 seed Sara Errani, 63 63. Bouchard's win over the World No.10 was the third Top 10 win of her career, her first two coming last year - she beat Sam Stosur at Charleston and Jelena Jankovic at Tokyo [Pan Pacific].
Halep and Bouchard will be playing against each other for the first time.
Also among Sunday's early winners was No.22 seed Alizé Cornet, who scored a heroic victory against No.14 seed Carla Suárez Navarro, 67(4) 75 63. Not only did the match last three hours and 26 minutes - the second-longest match of the year after Maria Sharapova's three-hour, 28-minute win over Karin Knapp at the Australian Open - but Cornet fought off triple match point serving 4-5, 0-40 second set.