Rybakina came into the clash the fresher of the two players after her second-round opponent Ajla Tomljanovic pulled out, but the Indian Wells and Rome champion, who had strapping on her right shoulder, made a sluggish start.
Cirstea broke the former Wimbledon champion in the fourth game of the first set and the seasoned world number 30 continued to dictate points from the centre of the court while taking an early lead in the contest.
The Romanian, who had lost her previous two meetings with Moscow-born Kazakh Rybakina, pounced early in the second set but dropped serve immediately as her opponent found her range again with some blistering groundstrokes.
Australian Open runner-up Rybakina failed to make the most of two set points at 6-5 and squandered two more chances to draw level in the tiebreak but got across the line with a cross-court winner, much to the joy of the Louis Armstrong Stadium crowd.
But Cirstea composed herself to take charge of the decider and went up 4-1 before staving off a late comeback attempt by Rybakina and closing out the match with a decisive late break after a double fault from her opponent.
"I've been playing for quite a few years and this was the best crowd I've ever played in front of," Cirstea said. "Thank you for the support, it's 12 o'clock so thank you for hanging out and making this amazing atmosphere.
"I'm very happy with the win today. I've lost twice against her and she's a good player, number four in the world and I knew it was going to be a tough battle. I'm just so happy, it's a great moment for me."
Up next for Cirstea is a clash with Swiss 15th seed Belinda Bencic but the Romanian said she was not looking too far ahead.
"It's one match at a time," Cirstea said. "At this stage in the tournament, everyone plays great and I'm glad that the work is paying off.
"Never stop believing in the things you do and the work you do because sooner or later if you put in the right energy... the things you want in life will come to you."
