Anisimova suffered a humbling 6-0, 6-0 defeat by Iga Swiatek in the title clash 12 months ago but the American showed no sign of any lingering scars as she swept aside her North Macedonian opponent on Court Two.
Gjorcheska, the first player from her nation to compete in a Grand Slam main draw, showed resistance towards the end in a bid to prolong the contest but Anisimova was too good for her and prevailed in 61 minutes.
"It's not easy playing a qualifier who has had a few matches under her belt," said Anisimova, who has struggled with a wrist injury in recent weeks.
"She played great, we haven't faced each other, so it was a tricky first round. I'm super excited to be back here.
"When I got here I was filled with great memories and I'm trying to carry that vibe ... trying to enjoy each day here."
Anisimova takes on Petra Marcinko or Sofia Kenin in the second round.