Dart also said she had numerous family supporters dotted around New York.
“My uncle and cousins live here, one of my cousins from the UK is over at the moment, my mum has been here since I started in qualifying, and my dad flew in last night,” she said.
“New York always feels like a home away from home.
“It feels like a community for me and I really felt the support out there.”
Dart could not convert any of her four break points in the opening game of the match, with Stearns not offering another opportunity in the first set.
Stearns piled relentless pressure on the Briton’s serve, eventually breaking in the eighth game.
Dart was broken for a second time in the opening game of the second set but, aided by an increasingly erratic Stearns, whose 40 winners were offset by 47 unforced errors, Dart secured two breaks of her own to force a decider.
Seizing control midway through the third set, Dart completed a crucial hold in the sixth game to end a run of three consecutive breaks before converting her fourth match point to seal a nervy win in a little under two and a half hours.
Dart joins compatriots Toby Samuel and Katie Boulter in the second round.