The American world number 116, who stunned Wimbledon champion Simona Halep in round three, beat Cirstea 7-5 6-2.
Townsend will face Canadian 15th seed Bianca Andreescu in the fourth round.
Andreescu defeated Caroline Wozniacki, the 19th seed, 6-4 6-4.
The 19-year-old Andreescu looks a real threat at Flushing Meadows this year and Wozniacki, twice a US Open runner-up, said: "She's playing very well and mixing up the pace. She has a lot of shots in her game that she can pull out.
"I think the one that I can most compare her game to would probably be Kim Clijsters, back in the day.
"I think because she moves well and she can stretch out and get to some balls and also play aggressive and using the angles.
"Everyone has their own style but for me, playing her, it a little bit brings me back to when I played Kim."
Germany's Julia Goerges pulled off a surprise by knocking out seventh seed Kiki Bertens.
Goerges, seeded 26, needed five match points but eventually got over the line 6-2 6-3.
American wild card Kim Ahn beat the former French Open champion Jelena Ostapenko 6-3 7-5.