Cirstea, who had taken out Errani's doubles partner - Roberta Vinci - in the first round, came out firing and, after dropping the second set, reeled off five games in a row to close it out, 62 57 61.
"Every single time against her I would come close but never get the win, but tonight I kept fighting and playing aggressive until the end," Cirstea said. "I never changed my game plan, no matter what the score was. I'm happy because I kept doing that until the end, and that's what gave me the win."
Cirstea scored the 13th Top 10 win of her career and Suárez Navarro her ninth Top 10 win.
Earlier in the day a few of the other favorites kept quietly moving through the draw, beginning with No.6 seed Caroline Wozniacki, who rallied from a break down in both sets to beat Annika Beck, 64 64.
"She's a fighter. She gets a lot of balls back. She mixes the rhythm and the pace of the ball, and I wasn't really hitting through the ball at times," Wozniacki said. "But I'm just happy to be through.
"It's such a difficult tournament. It's not like you get an easy one. Other tournaments you have a chance to get into the tournament, but here you really have to be ready from the start. It's like a sprint."
"It's always so difficult to play against a left-hander like Safarova - we had a tough match in Beijing, 6-4 in the third," Jankovic said afterwards. "She gives me a hard time, especially that lefty serve. I have a hard time breaking. It was a matter of who broke first. So my goal was to stay focused on my serve and try to hold it as much as possible and then try to break her serve when I got the opportunity.
"I lost my focus a little bit in the second set, but I stayed strong and pulled the match off."
Jankovic has a particularly tough task next - she plays Serena Williams in the quarterfinals.
"It's like every close ball, you pray, 'Call it, please!'" she joked. "It's exciting to be back on center court and to be playing against Serena Williams. I look forward to it. We'll see now how it goes.
"I have a lot of fans here and I love playing here, so we'll see what happens."
Williams has beaten Jankovic in all four of their meetings since she came back from her year-long lay-off, but two of those were three-setters and before that they were 4-4 (see the full H2H here).