The American teenager took just over an hour to dispatch her Ukrainian rival, who was a semi-finalist at both Wimbledon and Roland Garros last season.
"I love playing here, I love this stadium and the crowd is just great, so I'm glad I get to see everyone again in the next round," said Anisimova after registering her first win over a top 10 rival since the French Open last May.
Anisimova had to save two break points in the third game of the first set before holding. She then broke in the next game.
The World No.29 was under similar pressure in the second set but held on.
In the next round she will play either Svetlana Kuznetsova or and Polish teenager Iga Swiatek.
Recent finalist at Dubai, Elena Rybakina came from a set down against Sorana Cirstea to win 3-6, 6-3, 6-1 in a first-round match.
"Hopefully I can keep making finals and finally win another," the 20-year-old Kazakh said after her win.
Rybakina twice fell a break of serve behind before the Romanian closed out the opening set.
Her thunderous backhand swung the match and she will meet Belgium's Alison Van Uytvanck in the next round.
A string of lower seeds fell in the first round on Monday.
American Allison Riske, seeded 13, fell to compatriot Jennifer Brady 7-6 (12/10), 1-6, 7-6 (7/3) in a match that lasted 2 hours and 47 minutes.
Croatian Petra Martic, a semifinalist in Dubai last week and seeded 10, lost 4-6, 6-1, 6-2 to Czech Barbora Strycova. Croatian 17th seed Donna Vekic lost 6-4, 7-5 to Pole Iga Swiatek.
Marketa Vondrousova, a Czech seeded 12, won the first set but the subsided 1-6, 6-3, 6-2 to Saisai Cheng of China.