The paths of these two stars have intersected on several occasions due to their association with the luxury brand Bulgari
In the first match to be played from start to finish under the closed roof, Romania's Halep triumphed 6-3, 6-4.
She showed a flash of frustration after letting an early break get away in the opening frame, but she won the last three games to pocket the set and sealed the win when Safarova double-faulted on match point - one of 11 breaks of serve in the contest.
Halep apologized for her racquet-swiping fit of pique.
"I know it's not nice," she said. "But I can't control my emotions. I'm trying, but I can't do it."
Halep, who shot to prominence with a run to the 2014 French Open final, reached the semi-finals at Flushing Meadows last year, losing to eventual champion Flavia Pennetta.
She hasn't been much of a factor at this year's three prior Slams, making the quarter-finals at Wimbledon, the fourth round at Roland Garros and failing to win a match at the Australian Open.
But she said her confidence is high after a strong hardcourt build-up that included a victory in Montreal and she enjoyed the experience of playing under the new $150 million roof - which was closed as morning rain delayed the start of play on outside courts for an hour.
"It was amazing to have this experience," said Halep, although the closed roof didn't relieve all climate problems.
"I was a little cold," she said. "I felt the air conditioning."
Second-seeded Briton Andy Murray, riding a wave of confidence after claiming his third Grand Slam title at Wimbledon and a second Olympic gold medal in Rio, said he'd welcome playing under the roof when he takes on Spain's Marcel Granollers for a third round berth.
Murray has won six of seven encounters with Granollers, most recently beating him at the Indian Wells hardcourt Masters in March.
Serena Williams will be aiming to step it up against Vania King as she eyes a third-round berth.
World number one Williams, who matched the Open Era record for Grand Slam titles with her 22nd at Wimbledon, said she had plenty of room for improvement after a 6-3, 6-3 first-round win over Ekaterina Makarova.
"I think I can get a lot better," said the US superstar, who fired 12 aces and 27 winners against the Russian lefty - and said she'd have to wait and see if the work reignited the painful inflammation in her right shoulder.
Even if it does flare up, Williams has to be considered a strong favorite against her 87th-ranked compatriot King, a wild card she trounced 6-1, 6-0 in their only prior meeting, in the second round of the 2014 US Open.
Other marquee players eyeing a place in the last 32 on Thursday include Serena's elder sister Venus, the sixth, seed, who takes on German Julia Goerges.
Women's fourth seed Agnieszka Radwanska of Poland takes on Briton Naomi Broady.
Men's third seed Stan Wawrinka of Switzerland faces Italian qualifier Alessandro Giannessi in the first meeting between the two.
Sixth-seeded Kei Nishikori of Japan and eighth-seeded Dominic Thiem of Austria also face unfamiliar foes, Nishikori against Russian qualifier Karen Khachanov and Thiem against Lithuanian Ricardas Berankis.
The paths of these two stars have intersected on several occasions due to their association with the luxury brand Bulgari
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