The Canadian secured her place in the third round at Flushing Meadows for the first time in her short career but was made to work by Cirstea before eventually prevailing 6-2 6-7 6-4.
Bouchard will now face Czech 30th seed Barbora Zahlavova Strycova and admits she has some improving to do if she wants to go deeper in the tournament.
"I didn't feel as sharp as I want to be," she said.
"I was happy I could still play pretty solid but I didn't feel like I was at my highest level and she started playing very solid in the second and getting a lot of balls back and at the end of the day it was a fight and a battle.
"I definitely hope I can improve from this match because I am definitely not totally satisfied."
It was the first time the 20-year-old had played on the Arthur Ashe court and she insists she was not overawed playing on the biggest court on the professional circuit.
"It was cool, it was my first time on Ashe and it was special," she said.
"It was really cool that they played 'Genie in a Bottle' during the warm-up and I was like it is going well so far and this is pretty cool.
"They played it again at the end of the match and I felt pretty cool."