The Polish star is chasing a third grand slam title, and her second of the year after triumphing at the French Open in June.
Having held top spot in the rankings since April in the wake of Ash Barty's retirement, the 21-year-old is chasing her best result yet in New York, having reached the fourth round last year.
At Louis Armstrong Stadium on Tuesday she made an impressive start to her quest, with a near-flawless performance that saw her break Paolini seven times.
Swiatek showcased her ruthless streak barely a handful of minutes into the first set, rifling home three unanswered points on her first serve before promptly breaking the Italian for the first time.
Another break followed, before Paolini hit back to make matters 4-2, the only time she looked to seriously threaten the Pole's dominance.
Swiatek responded with her third break in a row, before bouncing back from another lost service game to seal the first set.
With momentum on her side, the Roland Garros victor bulldozed her opponent across the second set - and though Paolini was able to save match point once, there was never any doubt who was heading through to round two.
Swiatek is likely to face a more difficult challenge in the next round when she runs into 2017 champion Sloane Stephens, who battled back to see off Belgium's Greet Minnen 1-6 6-3 6-3 to the delight of the New York fans.
"I was just fighting as hard as I can and with your guys support, you pushed me through, so thank you," she said.
"Happy to get another win at my favourite tournament."
Joining Swiatek in the second round are American eighth seed Jessica Pegula, who beat Viktorija Golubic 6-2 6-2, and sixth-seeded Belarusian Aryna Sabalenka after her 6-1 6-3 win over Catherine Harrison.
Garbiñe Muguruza, seeded ninth, beat Clara Tauson 6-3 7-6 (7-5).
No. 22 seed Karolina Pliskova, the 2016 US Open runner-up, needed a third-set tiebreaker to edge Magda Linette, but No. 16 Jelena Ostapenko was ousted by China's Zheng Qinwen.
With agencies
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