Sharapova, who is seeded 22nd on her first appearance at Flushing Meadows since 2016, carved out a 3-0 first set lead despite making several mistakes early on and finished strongly with a dominant performance from the baseline.
Cirstea fought back with some hard hitting of her own, breaking the Sharapova serve in the first game of the second set and opening up a 4-2 lead before cracking under relentless pressure from the 2006 champion.
The Romanian sent a forehand into the net to offer Sharapova a break point, and allowed the Russian to level at 4-4 with a double fault.
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Both players exchanged further breaks of serve until Sharapova finally held to go 6-5 up, before piling pressure on her opponent as she served to stay in the match.
Cirstea committed her fourth double fault to give the Russian two match points and conceded the contest with her 32nd unforced error, which Sharapova greeted with a fist-pumping celebration.
The former world number one, who beat Swiss veteran Patty Schnyder in the first round, has never lost a night session match at Flushing Meadows in 22 attempts.
"Well, numbers are all relative, it's really a nice accomplishment but it doesn't win me any more Grand Slams," Sharapova said courtside as the clock ticked past 1:00 a.m.
"There's something about this court, something about New York, there's something about this crowd. You have to be a little crazy to be here past midnight, so thank you."
Sharapova will resume her quest for a sixth Grand Slam crown against 10th seed Ostapenko, who beat American Taylor Townsend 4-6 6-3 6-4 earlier in the day.
"It only gets tougher from here," the Russian added. "Being seeded in the 20s means you're going to get grand slam winners in the third round, as I have. But I'm looking forward to it."
(Reporting by Simon Jennings; Editing by John O'Brien)
