Sharapova, 30, was playing her first Grand Slam match since serving a 15-month doping suspension.
The 2006 champion and former world number one needed a wildcard to enter the main draw as she is ranked 146th.
When her name was drawn alongside Halep's in the first round it set up a contest that felt more like a Grand Slam decider than opener.
Over the course of two hours and 44 minutes, Sharapova's greater pace off the racquet outstripped Halep's speed of foot, with 60 winners proving decisive despite 64 errors.
"You never know what you're going to feel until you win that match point, but everything you go through is worth it for this moment," said Sharapova, who now has a 7-0 record against Romania's Halep.
"You sometimes wonder why you put in all the work, and this is exactly why."
Sharapova goes on to face Hungary's Timea Babos, the world number 59, in round two on Wednesday.
New York gives Sharapova warm welcome
Sharapova returned from her ban in April but has struggled with thigh and forearm injuries, playing just once since May, and was also refused a wildcard by the French Open.
The United States Tennis Association took a different view, handing the Russian a place in the main draw in New York, and she took full advantage.
John McEnroe had been among those wondering whether the Flushing Meadows crowd would react badly to the 2006 champion, but she was given a warm reception as she stepped on court.
It was Sharapova's first appearance at a Grand Slam since she lost to Serena Williams in the 2016 Australian Open quarter-finals - and she relished returning to the big stage.
"It's prime-time, baby - I love it," she joked afterwards. "I embrace every moment of it and I love it."
Halep has three times been on the brink of the number one ranking this year, and was match-tight after playing 12 times on North American hard courts over the last month, but in the end she was overpowered by Sharapova.
The five-time Grand Slam champion edged ahead in a gripping first four games that lasted 27 minutes, and a blizzard of winners brought her the first set.
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