Anisimova, who last beat a top-10 opponent on the way to her first Grand Slam semi-final at the French Open last year, struck 27 winners in a display of relentless aggression and saved all three break points on her own serve.
Svitolina, who finished the 2019 season as runner-up to Ash Barty at the WTA Finals, has struggled this year, having reached just one quarter-final at the Thailand Open after a third-round Australian Open exit.
The Ukrainian struggled to land her first serve during most of the contest and her poor shot selection ultimately proved her undoing.
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Anisimova, 18, will next meet either former world number two Svetlana Kuznetsova or Polish teenager Iga Swiatek.
"I think the most important thing was just staying focused and clearing my head, just not putting too much pressure on myself and just trying to enjoy (it)," Anisimova said.
"So I'm really happy to be back and kind of playing my own tennis again."
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Greece's Maria Sakkari celebrated breaking into the top 20 for the first time by defeating German Julia Goerges 6-4 6-3.
Sakkari, seeded 15th, won the opening four games in each set on her way to the second round for the first time in Doha after three attempts.
"She's got a big serve and a big game. You have to be careful with her and you have to come up with a good game plan to beat her," the 24-year-old Sakkari said.
"It's a dream come true to be in the Top 20 players. But I don't want to stop here. I see myself much higher and that's what I'm going to work for. It's a good goal, but I need to work more."
Sakkari next plays Czech Tereza Martincova who beat Japan's Misaki Doi in three sets.
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