The American eighth seed is chasing another shot at Grand Slam glory, building on a strong season that included her maiden WTA 1000 title in Doha and a run to the Wimbledon final, where she suffered a crushing 6-0 6-0 defeat to Iga Swiatek.
"It's not easy playing in the first round of a Grand Slam, I never played against her so it was tricky," Anisimova said.
"It's been a special year, I'm just enjoying the journey. I try to have fun every day but these are the moments I enjoy the most.
"I lost in the first round last year, so I'm really happy to have done better than that."
The 23-year-old, who returned to action last year after a long mental health break, was barely troubled in a one-sided affair at Flushing Meadows’ Louis Armstrong Stadium.
She broke to love for a 4–2 lead, and although the world number 83 broke back, Anisimova immediately reasserted control with another love break before serving out the first set with an ace.
Riding her momentum into the second, Anisimova broke Birrell twice, held serve comfortably to wrap up the win and advance to the second round. She will next face another Australian, Maya Joint.
Anisimova later said she was embracing the present moment in New York, refusing to dwell on the painful defeat she suffered in her first Grand Slam final at Wimbledon.
The world number nine said she was channelling her current form and Top-10 ranking into confidence rather than added pressure.
"If anything, I just embrace where I am and where my ranking is at when I step onto the court and just try and use it to my advantage," she told reporters.
Reflecting on her Wimbledon run, Anisimova believes those experiences will be invaluable in her home Grand Slam.
"I was playing some great tennis there, and there were a lot of tough matches that I got through," she said. "There were a lot of times where I dug really deep, and I think that’s going to be really important for me here."