The 87th-ranked American became the first player born this century to reach the fourth round of a Slam and the youngest American to do since Serena Williams did it at the 1998 French Open.
Anisimova takes first set with impressive display
Anisimova took charge right away, earning a break point in the opening game of the match after a successful challenge of a Sabalenka backhand and she converted when she blasted her own backhand past the Belarussian.
Tested on her own serve in the middle part of the set, the American came through with some relentless groundstrokes, often rendering the Belarussian helpless, wildly missing from all areas of the court.
The sixth game saw Anisimova fight off three deuces, eventually holding serve with a pair of aces for a 4-2 lead. After another confident hold, the American claimed the opening set on her third chance with a second break after a succession of piercing shots.
Teenager blows away Sabalenka to wrap up easy victory
If there were any nerves for the teenager, they weren't showing, holding to 15 to open the second set and again attacking Sabalenka, breaking the 11th seed after smoking a backhand up the line for a 2-0 lead.
Both players saved break points in their next service games and after Anisimova registered a love hold for a 4-1 lead, she went to work on finishing the Shenzhen champion off.
At 0-40 and staring into the abyss, Sabalenka pulled through the danger and held for 4-2. She couldn't make an impact on the Anisimova serve as the teenager again confidently held to move within a game of a stunning win.
Falling behind 30-0, the American again turned to her greatest weapon: her backhand as she created seemingly impossible angles to pull Sabalenka all over the court and on match point, the Belarussian flew a forehand long to seal a performance few will soon forget.
Anisimova speaks to the press
"She's a top player; she's had such a great year. I was just playing really well and she was playing really well, too", said Anisimova. "We had a great match and it just ended up going my way today".
Up next for the teenager is eighth seed Petra Kvitova, whom Anisimova defeated last year in Indian Wells. "Every player here is going to be great and every round here is going to be tough. I just focus on myself and I'm not really scared of anyone.
"I don't fear anyone. I just have a lot of confidence in myself, and I'm ready for tough matches. I believe in myself so much more than I ever did in this tournament. It's just trusting myself. That's why I have also been playing really well.
"I just want to win this tournament...right now", she said, smiling.